User:Hesselp/sandbox

Who prefers the current intro?

Having considered the comments on earlier proposals, I wrote a partly new version of the intro. Based on principles as:
- Considerable shortening
- As long is the title is not changed into "Function exp", the intro should start with a general definition applicable to all types of functions that are called 'exponential' by scientists. (See intro 5 Sep. 2009 – 25 Aug. 2015; see Talk 2016)
- The natural exponential function should have a prominent place.
- No mentioning of alternative definitions as power series, limiet of a sequence (it's not an infinite product !), continued fraction.
- Nothing about history, matrices, Lie algebra, Rudin, the natural exponential (=?), initial value problem, Riemannian manifold, bijection.
- Encyclopedic style, not a written classroom presentation.

Alternative intro 11-06
An exponential function, represented by written forms as  $a\,b^{x}$, $b^{x}$, $e^{x}$, $\exp x$ and more, is a mathematical function defined by the condition that pairs with the same difference, are transformed into pairs with the same ratio.  In math language:$\,f(u{+}d)/(f(u)=f(v{+}d)/f(v)$ for all $u$, $v$, $d$ .  An equivalent condition is: $f(x)/f'(x)$ is independent of $x$.  The $x$-independent ratio $f(x{+}1)/f(x)$ is called its base.

The unique exponential function ($\mathbb {R}$ to $\mathbb {R} ^{+}$) obeying  $f$(0) $=1$  and   $f(x)/f'(x)=1$ for all $x$,  is called  'the exponential function'  or  'the natural exponential function'. Symbol $\exp$, written as  $\exp x$ or $e^{x}$ or $e$^$x$.  Having Euler’s number e = 2.71828… as its base.

Exponential functions of type $f(x)=b^{x}$ obey $f$(0) $=1$  and   $f(x{+}y)=f(x)f(y)$ for all $x$, $y$ .  They transform addition into multiplication, the opposite of the main property of logarithmic functions.

Exponential functions of the general type  $f(x)=a\,b^{x}$ ($\mathbb {R}$ to $\mathbb {R} ^{+}$)  can illustrate two different meanings of the word 'base':  the exponential function  $f(x)=4\cdot 3^{2x}$ has base $f(x{+}1)/f(x)=\,9$,  while the expression  $3^{2x}$  has base $3$ (and exponent $2x$) .

Extending the (co-)domain to complex numbers leads to the complex exponential function $z\to e^{z}$, see § Complex plane
Extending the (co-)domain to quantities, exponential relations between quantities (in other sciences than pure mathematics) can be described. To avoid exponentiations with an invalid exponent, this functions are noted as $t\to a\,b^{t/\tau }$, with $b$ real positive (≠1), and $\tau$ the unit that measures quantity $t$ (mostly 'time'). See Exponential growth and Exponential decay.   [End of proposed intro.]

Who prefers the current intro? ~~

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Proposal for the lead, 3 Nov.'24

An exponential function, represented by written forms as $a\,b^{x}$, $b^{x}$, $e^{x}$, $\exp(x)$ and more, is a function defined by the condition that pairs of elements with the same difference in the domain, are transformed into pairs with the same ratio in the codomain  In math language:$\,f(u{+}d)/(f(u)=f(v{+}d)/f(v)$ for all domain elements u, v, d .  An equivalent condition is: $f(x)/f'(x)$ is independent of $x$.  The $x$-independent ratio $f(x{+}1)/f(x)$ is called its base.

The unique exponential function (R to R) obeying f(0)=1 and also f(x)/f'(x)=1 for all x, is called 'the exponential function', or 'the natural exponential function', written as $\exp(x)$ or $e^{x}$ or $e$^$x$. Having Euler’s number e = 2.71828… as its base (the x-indepent value of exp(x+1)/exp(x) )

Exponential functions of type $f(x)=b^{x}$ obey f(0)=1 and also $f(x{+}y)=f(x)f(y)$ for all $x$, $y$;  transforming addition into multiplication, the opposit of the main property of logarithmic functions.

Extending the (co-)domain to complex numbers results in the (complex) exponential function z -> e^z, see section ...
And extending to quantities Exponential relations between quantities (in other sciences than mathematics) can be described by forms as... See Exponential growth and Exponential decay. Unfinished. ++++++++++++++++++++++++++++++++++++

@Jacobolus: Thanks for your reaction. But . . . I don’t find answers on any of my three questions.
- Visiting linear function, power function or logarithmic function I don’t see a focus on one special individual function. So why the article exponential function s h o u l d  focus on $e^{x}$ ?
- “more general functions than function e x p”. See my first ‘remark’ (in Talk 31 Oct).
- The current text defines function $\exp$ in sentence nr. ??. That means a stronger ‘focus’ than sentence nr. 6 (‘’Most special type’’) in my proposal?
On your (2): start an article titled "Function $\exp$".  (3): "foo is exp of bar" ??  (4)Your "also" implies that type $a\,b^{x}$ is discussed in the current article/intro as well. So why it should be "confusing and overcomplicating" in my proposed intro?   Starting with type $a\,b^{x}$ avoids the undefined ídea of 'generalizing a function'. ~~

Intro reduced to essentials - proposal

An exponential function is a function obeying:  pairs of elements with the same difference in the domain, are transformed into pairs with the same ratio in the codomain  ($\,f(u{+}d)/(f(u)=f(v{+}d)/f(v)$ for all domain elements u, v, d ).  An equivalent condition is: $f(x)/f'(x)$ is independent of $x$.
The $x$-independent ratio $f(x{+}1)/f(x)$ is called its base.
Notation.  Exponential functions are usually written $a\ b^{x}$, using the notation of the two-variable function exponentiation.
Special type.  With a equals 1 ( f(0)=1), the functions $f(x)=b^{x}$ obey $f(x{+}y)=f(x)f(y)$ for all $x$, $y$  (transforming addition into multiplication, the opposit of the main property of logarithmic functions).
Most special type.  There is exactly one function obeying moreover $f(x)/f'(x)=1$ for all $x$, having Euler’s number e (2.71828…) as its base.  Usually named 'the exponential function' or 'the natural exponential function'.  Symbol: $\exp$,  written as  $\exp \,$(x),  e^x or  e^x . [End of poposed intro.]

Nine remarks:
- The order of the three types of exponential function from special to general is less easy to grisp than the order from general to special. Because: what is meant by "generalizations of a given function' (see hatnote), or with 'modification of a given function' (see Magyar25, 25 Oct.)? Why aren't 'generalizations/modifications' as well:  $e^{x^{2}}$,  $e^{[x]}$,  $x\,e^{x}$,  $e^{sinx}$,  $t\mapsto a\,e^{t/\tau }$ (time t to quantity ...),
- All info about quantity-to-quantity relations/functions ("applied settings") in a separate section, or in Exponential growth and Exponential decay.
- Not starting with how (a special case of) an exponential function is displayed in written form, but with its defining property.
- No history in the intro ("the exponential function originated from").
- "relating exponential functions to the elementary notion of exponentation". Not the exponential function 'an sich', but the usual way they are notated is related with exponentation.
- Don't use "base" (of an exponential function) without defining the term.
- No emphasis on b^x = e^{x ln b}. Every positive number, so every value of b^x as well, can be written as an exponentiation with an arbitrary (pos.) base. So with base e as well. Belongs to article Exponentiation.
- Not: "its rate of change at each point is proportional to the value of the function at that point."  Two numbers have a ratio, but two numbers cannot be proportional.
- Postpone (or avoid):     complex arguments,     matrices,     Lie-algebras,     symbol “ln” (not simple enough for the intro),     "the natural exponential" (= the natural exponentiation? sources),     the notation a b^kx (because b^k  instead of a single variable, suggests that k stands for a quantity instead of a number),     "initial value problem",     power series definition,     square matrices,     Lie groups,     Riemannian manifold,     antilogarithm (is/was used as well for the inverse of logarithmic functions with base 10 and other bases).

Comments? ~~

Natural / general / more general exponential function

@Alsosaid1987, Magyar25, and 91.170.28.20: In the introduction the word ‘exponential’ seems to be used in four ways:
- The exponential function ($e^{x}$)
- also known as exponential functions ($b^{x}$)
- allows general exponential functions (?)
- more generally also known as exponential functions ($ce^{ax}=cb^{kx}$)
So well defined names, sometimes with alternatives, are (imho) strongly desired.
My question:  Who knows something better (at least for use in this article) than:
- General exponential function(s)   ($ab^{x}$)
- Special exponential function(s) / Exponential function(s) / Zero-to-one (f(0)=1) exponential function(s)   ($b^{x}$)
- The natural exponential function / The exponential function   ($e^{x}$)

Similar names
The nomenclature described above could be extended to:
- General logarithmic function(s)   ($a\log _{b}x$)
- Special logarithmic function(s) / Logarithmic function(s) / One-to-zero logarithmic function(s)   ($\log _{b}x$)
- The natural logarithmic function / The logarithmic function   ($\ln x$)
- General power function(s)   ($ax^{d}$)
- Special power function(s) / Power function(s) / One-to-one power function(s)  ($x^{d}$).

Extension to quantities
The variables in the notations of the functions discussed above are meant as reals.
Exponential growth and decay can be described by functions with quantities as variables. E.g. written as:  $x\mapsto a\,b^{x/s}$, b real >1   and   $x\mapsto a\,b^{x/s}$, 0<b<1   with x, a and s quantities, x and s of the same kind.
Names of this functions: “Exponential growth” and “Exponential decay”.

The general exponential functions, as well as the functions ‘exponential growth’ and ‘exponential decay’, comply with   $f(x+s)/f(x)=f(y+s)/f(y)$  for all x, y, s (all reals or all quantities of the same kind).  In words: this functions transform equidistant pairs into equiratio pairs.

The first sentence is wrong - since 2015

Exponentiation is the name (one of the names?) of an operation R⁺×R→R, 'involving two numbers'. So its inverse is not a function at all.
I suppose that was meant: a logarithm is a (written) expression with a symbol for a certain logarithmic function, combined with a symbolic expression for a domain element.
A function f on R⁺ is called logarithmic when it transforms multiplication into addition: f(uv) = f(u) + f(v) . Or - in an indirect way - when it is the inverse function of an exponential function of type x→b^x (b>0, ≠1) .
Or anything equivalent.   Yes?   ~~

This seems excessively pedantic and confusing, but it's plausible we could make up a better first few paragraphs. If you want to be precise, the logarithm function is the inverse of the exponential function (or a logarithm function is the inverse of an exponential function). The term logarithm, most precisely, is a synonym for exponent, but saying it that way can also be confusing to novices. The unadorned term logarithm is also routinely applied to the logarithm function, or to an expression such as ⁠ ln ⁡ x {\displaystyle \ln x}⁠ or ⁠ log b ⁡ a {\displaystyle \log _{b}a}⁠. –jacobolus (t) 20:13, 21 August 2024 (UTC)

{Jacobolus} 'Excessively pedantic' Really? The property of a function f (R⁺→R) to be called logarithmic is that it transforms an arbitrary multiplication into an addition: f(uv) = f(u) + f(v) . As standard name for a function of this kind, I prefer logaritmic function over logarithm function, because the adjective 'logarithmic' describes more directly its main property. And for the special spiral I've only seen 'logarithmic spiral'. Google hits: 577000 to 161000.   Can this be used in the lede? ~~

Defining the logarithmic function

A logarithmic function (or logarithm function) f (R⁺→R) transforms multiplications into additions, that is: f(uv) = f(u) + f(v) for all pairs u, v of positive numbers.
Is this simpel and clear enough to start the lead with? ~~

That is a useful property of the logarithm function, but it does not describe what the function is. I don't see what's wrong with the current lead. Each exponential function $\exp _{b}(x)=b^{x}$ has an inverse logarithm function $\log _{b}(x)$. The log is the inverse of the exponential. The fact that there is a family of exponential functions, and a corresponding family of logarithm functions, is detailed later.

You could also define a different two-variable powering function $f(x,y)=x^{y}$ rather than considering a family of single-variable exponential functions but that does not have a well-defined inverse as it is not even locally 1-1 from its inputs (the pair $x,y$) to its outputs (a single number). —David Eppstein (talk) 19:27, 24 August 2024 (UTC)

An alternative would be to say something like: "In the equation ⁠$y=b^{x}$⁠ the quantity ⁠$b$⁠ is called the base, the quantity ⁠$x$⁠ is called the exponent, and the quantity ⁠$y$⁠ is called the result. When this equation is rewritten to isolate the exponent ⁠$x$⁠, ⁠$x$⁠ is instead called the logarithm base ⁠$b$⁠ of ⁠$y$⁠, denoted ⁠$x=\log _{b}y$⁠." But this is probably just as (if not more) confusing. –jacobolus (t) 19:44, 24 August 2024 (UTC)

"it does not describe what the function is" ??

@David Eppstein: Please can you motivate why (an - improved? - variant of Aug 24):
A function f (R⁺→R) is called logarithmic function (or logarithm function)   iff   f(uv) = f(u) + f(v) for all pairs u, v of positive numbers.
you don't see as a precise definition?
Is the definition precise, using the inverse function?:
A function f (R⁺→R) is called logarithmic function (or logarithm function)   iff   f^inv(u) + f^inv(v) = f^inv(uv) for all pairs u, v of positive numbers.
Or should it be?:
A function f (R⁺→R) is called logarithmic function (or logarithm function)   iff   b ^ f(x) = x,  b>0 ≠1, x>0 .
presuming knowledge of the not very elementary (IMHO) bivariate operation ^ for real variables.

@Jacobolus: Why "the quantity b, x, y" ? This letters are used here as variables. They all three represent real numbers, not more general quantities.
The definition of "logarithm base b of y" really needs a mysterious rewriting of an equation? There really isn't a more direct way?
And after all I don't see a definition of logarithmic function / logarithm function. ~~

- let it be so. And about 'confusing', your reaction (thanks for it) allows me to further clarify some points.

• the logarithm function is the inverse of the exponential function.   Do you mean: 'the natural logarithm function is the inverse of the natural exponential function'? Easier to interprete, not only for a beginner.
  
• a logarithm function is the inverse of an exponential function.   Not wrong, but very uninformative:
  

Somewhat better is 'a logarithm function is the inverse of an exponential function of type x→b^x (b>0, ≠1) .(For the inverse of a (general) exponential function, of the form x→a·b^(x/ξ), doesn't transform multiplication into addition.)

• Why use the name 'logarithm function' in stead of 'logarithmic function'? Confusing.

{Lpola} Your call for 'clear distinctions' brings me to the presentation of (not very well known?) verbal characterizations of four main types of functions, together with a subtype of each.

• A lineair function transforms equidistant pairs into equidistant pairs:   f(u+s) - f(u) = f(v+s) - f(v) .

    * A proportional(?) function transforms addition into addition:   f(u+v) = f(u) + f(v) .

• An exponential function transforms equidistant pairs into equiratio pairs:   f(u+s) / f(u) = f(v+s) / f(v) .

    * An anti-logarithmic(?) function (R→R⁺) transforms addition into multiplication:   f(u+v) = f(u) • f(v) .

• An anti-exponential(?) function transforms equiratio pairs into equidistant pairs:   f(ru) - f(u) = f(rv) - f(v) .

    * A logarithmic function (R⁺→R) transforms multiplication into addition:   f(uv) = f(u) + f(v) .

• A general power(?) function transforms equiratio pairs into equiratio pairs:   f(ru) / f(u) = f(rv) / f(v) .

    * A power(?) function (R⁺→R⁺) transforms multiplications into multiplications   f(uv) = f(u) · f(v) .

A much harder point (but important for you - and for me as well) is how to get the whole WPen accept unique names in all eight cases? Short, but very artificial and therefore chanceless, should be: s-s-functions, a-a-functions, s-r-functions, etc.

About sources. It's hardly to believe that this simple scheme shouldn't be ever mentioned on WPen. Or on other WPs, or other internet-pages. Half of the scheme - the four 'subtypes' - has a famous source, 200 years old: C-A Cauchy, Cours d'analyse 1821, Chap.V, pp. 103-122. So maybe the other four types could be found in books from that period? ~~

Alternative for: "It is the value at 1 of the (natural) exponential function"

The objection (by user 73.218.32.185, summary of edit 30 June 2024): “It is the value 1 of e^x”. Uhhh…obviously. Just as every constant raised to the first power is equal to the number you started with. This sentence is useless and should be removed. can be met by replacing the sentence by:
" It is the value of $\ (f(a)+f(b)/f'(b))\,/\,f(a)\$ for any general exponential function $f$ and any two numbers $a$ and $b$ . "

Alternative 2 (special case):   " It is, with $\ f(x)\equiv 2^{x}$,  the value at $0$ of $\ f(f/f')/f\$. "

Comments? ~~

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Generalization of definitions three and four (article section 'Definitions')

For any real function $f$ transforming equally spaced pairs into relatively equal pairs, and for any two numbers $u\,,\,v\ :$

$\quad e\ \equiv \ {\frac {f(u\,+\,f(v)/f'(v)\,)}{f(u)}}\ .$

A 'general exponential' function f is usually described as

$f(x)\ \ =\ \ a\,b^{x}\ \ =\ \ f(0)\ (f(1)/f(0))^{x}$ with a = f(0)

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On the roots of the universal numerical constant e

The definition of number e as being connected with a very special (the 'natural') exponential function, and with a very special (the 'natural') logarithmic function,

the value of parameter a giving the graph of the function x→a^x a nice 45^o slope at x=0 (and giving the graph of the function x→log_a x the same slope at x=1)

neglects the fact that this number (2,71828...) is connected with functions of a much broader type: all general exponential functions and all general logarithmic functions. This relation can be explained as follows.

The condition for a function f→f(x) ( R→R, or describing a changing quantity as function of another quantity (often: time) ) to be called exponential can be given in different ways - proofs of their equivalency are omitted:

• Function f transforms any equally spaced pairs of the independent variable (say (u, u+s), (v, v+s) ) into relatively equal pairs ( f(u), f(u+s) ), (f(v), f(v+s) ).   So f(u+s) / f(u) = f(v+s) / f(v) .
• The ratio of the slope of the tangent at a point P (p, f(p)) on the graph of f, and f(p), is independent of the position of P.   Formulated with derivation: f'(x) / f(x) doesn't depend on x.
• The distance between points (u,0), (v,0) on the x-axis, with (u,0) on the tangent to the graph of f at point P (v, f(v)), is independent of the position of P. This constant distance is called:the subtangent (τ, tau) of an exponential function.
• f(x) = a · b^(x/ξ),   a := f(0),   ξ an arbitrary unit for the independent variable,   b := f(x+ξ) / f(x) > 0 ;  the operation '^' (exponentiation) is supposed to be known.
• f(x) = a · e^(x/τ) ,   a := f(0),   τ := the constant subtangent (f/f') of f,   e := f(x+τ) / f(x)  the universal constant ratio of function values at any two domain points at subtangent distance, for any exponential function f.

Another point I don't see clearly how to motivate why the expression  a · b^(x/ξ)  should be more usefull than  a · e^(x/τ) .   People speak about the half time of U-235 (b=1/2), not its one-over-e-time.  And about doubling time.

I touched on this subject already seven years ago. See . . .

Bronnen

Sometimes the use of the label 'exponential' is restricted to real functions x->f(x) obeying:

• f(u+v) = f(u) · f(v) for every pair (u,v). (note Cauchy)
• f(x)/f'(x) = 1 for every x.
• the constant subtangent equals 1.
• f(x) = b^(x/ξ), ξ an arbitrary choosen unit for the independent variable, b=f(x+ξ)/f(x) > 0
• f(x) = b^x , b = f(x+1)/f(x) > 0 .

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Intro: Hele talkpage doorgenomen, starting at 2004, not missing the arguments in the RfC (March/April 2016), ook van drie verwante/neighbouring pages. Intensive study of the problems raised there, during .... Sources not mentioned here for the moment, so probably it will classified as POV.

(title) == Exponential function == (hatnote) Related articles: E (mathematical constant, Exponential growth, Exponential decay .

The word exponential is used in two different - related - ways.

• As an adjective, denoting a certain property of mathematical functions, and of changing quantities outside mathematics.

Grains on the chessboard, compound interest, kroosjes in de vijver, radioactive decay,

• As part of the names the natural exponential function or the exponential function denoting a specific function with a central role in mathematical analysis.

Same situation with the sine function as name, denoting a specifik (weel known) function, and sinusoidal as adjective denoting a certain property of mathematical functions. Now with two different words!

Comment on the undoing by Sapphorain, 2 April 2017

1. The original wording by Saidak can be of interest on a special Saidak-page, not on the page titled "Euclid's theorem".
2. I cannot see why the 7-lines version of the proof should be more accessible for laymen than the short version. For:
2a. The central argument in the proof is described in essentially the same way in both versions: "For any n > 1, n and n + 1 have no common factors; they are coprime" versus "Because all prime divisors of a natural number n are different from the prime divisors of n+1, . . . ".
2b. The short version doesn't have the "induction for any k".
2c. Nor the superfluous and difficult to read and interprete "example" at the end.
2d. Using N₂, N₃, ..., but not N₁, is confusing.
2e. Why 'at least' in '1806 has at least four different prime factors' ?
Who refutes this arguments? Who has a better proposal for the text of the proof? -- Hesselp (talk) 21:42, 5 April 2017 (UTC)

Second attempt to improve the wording

Since each natural number (≥2) has at least one prime factor, and two successive numbers n and (n+1) don't have any prime factor in common, the product n×(n+1) has more different prime factors than the number n itself.
This implies that each term in the infinite sequence:     1,   2 (1×2),   6 (2×3),   42 (6×7),   1806 (42×43),   (1806×1807),   (1806×1807) × (1806×1807 + 1),   · · ·    has more different prime factors than the preceding.
The sequence never ends, so the number of different primes never ceases to increase. --

===Second attempt plus === Improved wording of Saidak's proof; discussion in Talk.
Since each natural number (≥2) has at least one prime factor, and two successive numbers n and (n+1) don't have any factor in common, the product n×(n+1) has more different prime factors than the number n itself.  So in the chain of pronic numbers:
1×2 = 2 {2},    2×3 = 6 {2, 3},    6×7 = 42 {2,3, 7},    42×43 = 1806 {2,3,7, 43},    1806×1807 = 3263443 {2,3,7,43, 13,139}, · · ·
the number of different primes per term will increase forever. --

Elaborating Lazard's description of 'series' as an expression

I'm pleased to see that Lazard (Febr.14, 2017, line 4) describes the meaning of the word series as an expression of a certain type.   Less clear (or better: mysterious) is the remark: "obtained by adding together all terms of the associated sequence"; what could be meant by "adding together"? What kind of action should be performed, by who, on which occasion, to obtain / create an expression of the intended kind?
More remarks on the present text of the article, in this Talk page: 15:14 16 April 2017.
To get things clear, I propose to start this article in about the following way:

I n t r o d u c t i o n
In mathematics (calculus), the word series is primarily used for expressions of a certain kind, denoting numbers (or functions).
Symbolic forms like    $a_{1}+a_{2}+a_{3}+\cdots$    and    $\sum a$  or  $\sum _{n=1}^{\infty }a_{n}$   expressing a number as the limit of the partial sums of sequence $a$, are called series expression or shorter series.

Secondly, in a more abstract sense, series is used for a certain kind of representation (of a number or a function),  and also for a special type of such a series representation named series expansion (of a function, e.g. Maclaurin series, Fourier series).

And thirdly, series can be synonymous with sequence.  Cauchy defined the word series by "an infinite sequence of real numbers".[source: Cours d'Analyse, p.123, p.2, 1821, 2009]
This use of the word 'series' can be seen as somewhat outdated.

The study of series is a major part of mathematical analysis. Series are used in most areas of mathematics, even for studying finite structures (such as in combinatorics), through generating functions. In addition to their ubiquity in mathematics, infinite series are also widely used in other quantitative disciplines such as physics, computer science, statistics and finance.

C o n t e n t s

D e f i n i t i o n s,   c o m m o n  w o r d i n g s
Given a infinite sequence $a$ with terms $a_{1},a_{2},a_{3}$ et cetera (or starting with $a_{0}$) for which addition is defined, the sequence
$\quad a_{1},\quad a_{1}+a_{2},\quad a_{1}+a_{2}+a_{3},\ \ ...$     is called  the sequence of partial sums of sequence $a$ .
Alternative notation:     $(a_{1}+\cdots +a_{n})_{n=1,2,\cdots }$  .
Example: The sequence 1, 2, 3, 4, ···  is the sequence of partial sums of sequence 1, 1, 1, 1, ··· ;  the sequence 1, 1, 1, 1,···  is the sequence of partial sums of sequence 1, 0, 0, 0,··· ;  this can be extended in both directions.

A series is a written expression using mathematical signs, consisting of
- an expression denoting the function that maps a given sequence on the limit of its sequence of partial sums
combined with
- an expression denoting an infinite sequence (with addition and distance defined).

Second meaning   The symbolic forms   $a_{1}+a_{2}+a_{3}+\cdots$  (plusses-bullets form)   and   $\sum _{n\geq 1}a_{n}$  (capital-sigma form)
are sometimes used to denote the sequence of partial sums of sequence $a$ , instead of the value of its eventual existing limit.

A sequence is called summable iff its sequence of partial sums converges (has a finite limit, named: sum of the sequence).

Convergent / divergent series   The combination convergent series shouldn't be interpreted literally, for an expression itself cannot be convergent or divergent.  By tradition  "Σ $a$ is a convergent series"  as well as  "series Σ $a$ converges"  are used to express that sequence $a$ is summable.   Similarly, "Σ $a$ is a divergent series"  and  "series Σ $a$ diverges"  are used to say that sequence $a$ is not summable.

Convergence test for series   Again, this traditional wording cannot be taken literally because 'series' is the name of an expression of a certain kind, not the name of a mathematical notion. An alternative is: summability test for sequences.

Absolute convergent series   This is the traditional naming for a sequence with summable absolute values of its terms. The alternative absolute summable sequence is not in common use.

Series Σ $a$  and  sequence $a$  are interchangeable in traditional clauses like:
- the sum of series Σ $a$,   the terms of series Σ $a$,   the (sequence of) partial sums of series Σ $a$,
  the Cauchy product of series Σ $a$ and series Σ $b$
- the series Σ $a$ is geometric, arithmetic, harmonic, alternating, non negative, increasing  (and more).

There is no standard interpretation for the limit of series Σ $a$.

S e r i e s  r e p r e s e n t a t i o n   o f   n u m b e r s   a n d   f u n c t i o n s
In some contexts the word 'series' shouldn't be seen as referring to a certain type of written symbolic expressions, but as referring to a special type of representation of numbers (and functions). Namely: defining a (irrational) number as the limit of the partial sums of a known infinite sequence of (rational or irrational) numbers. And in the case of functions: defining a function as the limit of the partial sums of an infinite sequence of functions (which are seen as 'easier' or more elementary in one way or another than the function represented by the limit).
Examples of the use of the word 'series' in this sense, can be seen in the final sentences of the introduction above, starting with "The study of series is a major part ...".

As comparable with the idea of series representation or infinite sum representation can be seen:  the continued fraction representation and the infinite product representation (for numbers and functions).

S e r i e s  e x p a n s i o n   o f   f u n c t i o n s
The combination 'series expansion' is used for a special type of series representation of functions. ('Series expansion of numbers '  is meaningless.)
A series expansion is a representation of a function by means of the infinite sum of a sequence of power functions of increasing degree, in one of its variables. Or functions like (for example) $x\rightarrow a_{n}(x-b)^{n},\ \ x\rightarrow a_{n}\sin ^{n}x+b_{n}\cos ^{n}x$.
The labels Maclaurin series, Taylor series, Fourier series shouldn't be seen as denoting expressions but rather representations of the type series expansion. So Maclaurin series should be understood as Maclaurin expansion, Fourier series as Fourier expansion, et cetera. [Source: WolframMathWorld series expansion and Maclaurin series].

P o w e r  s e r i e s
"Power series" can be used
- as synonym for "Maclaurin expansion", and
- for a series expression which includes a sequence of power functions with increasing degree.

C a u c h y   a s   s o u r c e   o f   c o n f u s i o n
Cauchy, in his 'Cours d'Analyse' (1821) made an important, but quite subtile, distinction between the meaning of 'to converge' and 'being convergent':
- a sequence (French: suite) can converge (both French and English) to a limit, versus
- an infinite sequence of real numbers (named 'série' by Cauchy) having its sequence of partial sums converging to a limit, the first sequence named 'une série convergente ' .
Only a tiny difference between 'sequence' and 'series', but an essential one between 'converging' and 'convergent'.
This imprudent choise caused permanent confusion around the use of the word 'series'(e.g. in the German translations of 'Cours d'Analyse' of 1828 and 1885), until the present day.
[sources: Cauchy, see p.123 and p.2 quantité C.L.B. Susler, 1828, Susler, S.92, Carl Itzigsohn, 1885, Bradley/Sandifer, 2009 ]

[More sources on the problem with 'series' in books/publications by: professor H. Von Mangoldt, E.J. Dijksterhuis, H.B.A. Bockwinkel, professor N.G. de Bruijn, professor A.C.M. van Rooij, professor D.A. Quadling, Mike Spivack, H.N. Pot; links have to be added. Several of this sources are written in Dutch.]

Overwegen om delen van de rest van dit artikel onder te brengen op andere pagina's. Vaak staat er nu al een verwijzing naar "Main article".

Rekenregels Rewriting voor serie expressions

How to denote a sequence?

To 166.216.158.233, and ... .  On Februari 28 2017, you changed {s_k} into (s_k) at several places. I understand your argument (a sequence is a mapping, not a set), but I see your solution as insufficient. For without any harm, you can do without braces/parentheses at all, and without any index symbol as well.   A sequence is defined as a mapping on the set of naturals, so label them with a single letter. Just as people mostly do with mappings/functions with other sets as domain: f, g, F, G, ... .
When there is a risk of confusion you can write "sequence s", "sequence S"  in stead of just "s" or "S".
Who has objections? (Yes, I know the index is tradition, but it is superfluous and therefore disturbing.)
In the Definition section, three lines after "More generally ..." I read:
    the function $a:\mathbb {N} \mapsto G$ is a sequence denoted by $a(n)=a_{n}$.
I count three different notations for the same domain-$\mathbb {N}$-function (sequence), four lines later a fourth version - $(a_{n})$ - is used.
Last remark: It's not correct to say that sequences ($(s_{n})$ and $(a_{n})$) are subsets of semigroup $G$. --

Index sets as generalization (subsection Definition)

For me it is impossible to find any information in the second part of subsection 'Definition'- after 'More generally....'.
The text seems to suggest that the notion of "series" (whatever that is ...) can be extended from something associated with sequences (mappings on the set of naturals) to a comparable 'something' associated with mappings on more general index sets.  But nothing is said about how such generalized mappings $a$ can be transformed into a limit number $L$.  Is it possible to generalize the tric with the 'partial sums'? This index sets has to be countable? No reference is given. (The present text is composed by Chetrasho July 27, 2011).
I propose to skip the text from 'More generally' until 'Convergent series'.   Any objections? --

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My editing priorities: 1.Correct and non-misleading statements 2.Sufficient referencing 3.Accessible leads 4.Article structure

Hello Bill Cherowitzo.   You are right, my two questions are answered in the final section of the article.
But I persist that the description of the notion named series becomes even more unclear by adding six sentences (the greater part of the Definition section) on a generalization that will be unknown to most readers.
Moreover, the correlation between the position of this notion connected with sequences, and its position connected with mappings on an index set, is not very strong. For:
In (elementary) calculus two different symbolic forms (both named 'series') are used, expressing the relation between a sequence and its 'sum'. One of them, the plusses-bullets form $a_{1}+a_{2}+a_{3}+...$ cannot be used in the generalized situation. And the other one, the capital-sigma form needs adaption ($i\in I$ instead of $i=1,2,3,\cdots$ or $i=1\ \infty$ or $i\geq 1$ or $i\geq 0$).
The absence of relevant information in this six sentences is not undone by a 'lack of information tag'. Skipping this sentences I cannot see as a "removal of [relevant] material". --

Openingszin datum ???
Infinite series
The sum of an infinite series a0+a1+a2+... is the limit of the sequence of partial sums S N = a 0 + a 1 + a 2 + . . . + a N {\displaystyle S_{N}=a_{0}+a_{1}+a_{2}+...+a_{N}} {\displaystyle S_{N}=a_{0}+a_{1}+a_{2}+...+a_{N}} , as N → ∞. This limit can have a finite value; if it does, the series is said to converge; if it does not, it is said to diverge. The fact that infinite series can converge resolves several of Zeno's paradoxes.

Openingszin 30 March 2006 16:48
In mathematics, a series is often represented as the sum of a sequence of terms. That is, a series is represented as a list of numbers with addition operations between them, e.g,

1 + 2 + 3 + 4 + 5 + ...

which may or may not be meaningful, as it will be explained below.

Again on the Definition section

Yesterday's (13 April 2017) reduction in this section is an improvement, yes. Now this shorter version makes it easier to explain my objection to its central message. I paraphrase this message in the next four lines:
1. For any sequence $a$ is defined a
2. associated series Σ$a$ (defined as: an ordered "element of the free abelian group with a given set as basis" - the link says).
3. To series Σ$a$ is associated the
4. sequence $s$ of the partial sums of $a$ .
Why in line 2 an 3 a detour via a double 'association'(?) with something named 'series'? Is the meaning of that word clearly explained in this way to a reader? I don't think so.   I'm working on a text that starts with:
"In mathematics the word series is primarily used for expressions of a certain kind, denoting numbers (or functions). Secondly"
I plan to post this within a few days. -- Hesselp (talk) 13:39, 14 April 2017 (UTC).

Who can tell me how to find out whether or not a given "ordered element of the free abelian group with a given set as basis" has 100 as its sum? Who can mention a 'reliable source' where the answer can be found?
Why should this mysterious serieses be introduced at all, in a situation where it's completely clear what it means that a given sequence has 100 as its sum. I cannot find a motivation for this in a 'reliable source' mentioned in the present article.
So skip this humbug (excusez le mot).

About an eventual 'immediate removal': Should I have to expect that a majority in the Wiki community will support removing a serious attempt to describe in which way (ways!) the word 'series' is used in most existing mathematical texts. And replace a version including a 'definition' which has nothing to do with the way this word is used in practice; only because the wording has some resemblance with meaningless wordings that can be found in (yes, quite a lot of) textbooks.
In the present 'definition' of series the words 'formal sum' are linked to a text on Free abelian groups. Can this be seen as a 'reliable source' for a reader who wants to know what could be meant by 'formal sum'?  Wikipedia is not open for attempts to improve this? --

Comments on changes in the Definition section

Line 3, quotation: "Summation notation....to denote a series, ..."
A notation to denote an expression ?? Sounds strange (first sentence says: series = expression of certain kind).

Line 4, quotation: "Series are formal sums, meaning... by plus signs),"
I can read this as: "The word 'sum' has different meanings, but the combination 'formal sum' is a substitute for 'series' (being forms consisting of sequence elements/terms separated by plus signs)".   Correct?, this is what is meant?
But "Series are formal sums" seems to communicate not exactly the same as " 'series' is synonym with 'formal sum' ".

Line 4-bis, quotation: "these objects are defined in terms of their form"
With 'these objects' will be meant: 'these expressions (as shown in the first sentence)', I suppose. But then I miss the sense of this clause. An expression IS a form, and don't has to be defined (or described?) in terms OF its form.

Line 6. Properties of expressions? and operations defined on expressions? This regards operations as enlarging, or changing into bold face, or ...?

Line 7. "...convergence of a series". In other words: "convergence of a certain expression"?   I'm lost.

I'll show an alternative. --

=== Three proposals for adaptations in the Definition section ===
I. Note 3 in the present text, saying "...a more abstract definition....is given in....", should be removed.
For it doesn't have any sense to refer to a  'more abstract definition'  of
    an expression of the form   $a_{0}+a_{1}+a_{2}+\cdots$ ,  labeled with the name 'series'.
There is not a  'less abstract definition'  of this kind of expression either. Only a description.

II. A more direct formulation of the third sentence in this section is:
"A series is also called formal sum, for a series expression has a well-defined form with plus signs."

III. Remarks on the 'usefullness' and the 'fundamental propery' of such expressions of the form   $a_{0}+a_{1}+a_{2}+\cdots$ , shouldn't be included in a definition section. --

H o w   t o   r e d u c e   c o n f u s i o n
The best thing to do is:  Stop using the word 'series' at all, and say:
(absolute) summable sequence and summability tests,  in stead of: (absolute) convergent series and convergence tests .
Second best is: inform students and readers of Wikipedia about the historical source of the confusion. Let them understand that the existence of any definable notion 'series' (different from 'sequence') is a wide-spread misconception. And train them to interprete (absolute) convergent series as nothing else as summable sequence. --

Answer to Lazard: Thank you for contributing to the search for the best way to describe what is meant with the word 'series' in texts on mathematics(calculus). I saw some points in your rewriting of the Definition section which I can see as improvements. But there are some problems left:
1. Rewording the first sentence more close to the usual way as definition of 'infinite series / series',  I get:
An infinite sum is called series or infinite series if represented by an expression of the form: $a_{0}+a_{1}+a_{2}+\cdots ,$ where ...
This paraphrasing is correct?
Please add an explanation of what you mean by 'infinite sum'. &nbnp;And tell how a blind person can decide whether or not he is allowed to say 'series' to such an infinite sum, as he cannot see the form of the representation.
2. In the third sentence 'summation notation' is introduced, showing a 'capital-sigma' form, followed by an equal sign and a 'plusses-bullets' form. Why two different forms to illustrate the 'summation notation'?
3. Please explain what you mean with 'formal sum' (fourth sentence). See this discussion. And the same question for 'summation' at the end of that sentence.
4. Your seventh sentence end with "...the convergence of a series". Do you really mean to define "the convergence of an expression(of a certain type)?
5. Finally, I'ld like to see an explanation of the clause "the expression obtained by adding all those [an infinite number of] terms together" (fifth sentence in the intro). I don't see how the activity of 'adding' (of infinite many terms!) can have an 'expression' as result. --

R e d u c t i o n   o f   s u m s   a n d     p r o d u c t s

A sum of two numbers given in series representation,
a product of two numbers given in series representation, and
a product of two numbers, one of them given in series representation,
can be reduced according to:

$\sum$$_{i}\ a_{i}\ +\,\sum$$_{i}\ b_{i}\ =\,\sum$$_{i}\ (a_{i}+b_{i})$

$\sum$$_{i}\ a_{i}\ \times \,\sum$$_{i}\ b_{i}\ =\,\sum$$_{i}\ (a_{1}b_{i}+\cdots +a_{i}b_{1})$     ($(|a_{i}|)$ or $(|b_{i}|)$ summable)

$\sum$$_{i}\ a_{i}\ \times \quad \ c\quad \ \ \,=\,\sum$$_{i}\ (a_{i}\,c)$ .

The same applies for functions instead of numbers. --

Reaction to D.Lazard. On his remarks concerning my 'lacking understanding' of what a series IS, and my proposals for rewriting THE definition of a series. (The 'IS' and 'THE' referring to Lazard's personal POV.)

Cauchy used 'série' in his publications according to the definition:
    "On appelle série une suite indefinite de quantités (= nombres réelles)".  See 1821 Cours d'Analyse p.123,2
You agree that in modern English this reads as "An infinite sequence of reals is called series." ?  A clear definition?
(Maybe later on Cauchy used the same word to denote sequences of complex numbers as well.)
Probably by his choice for "convergente" naming the property now called "Fr: sommable / En: summable", a permanent confusion arose.
Numerous alternative attempts to define 'series' can be found, all of them denying Cauchy's distinction between 'converger / to converge'  versus  'convergente / convergent'. This attempts can be quite diverse, see for instance Bourbaki's: "a pair of sequences (a_n), (s_n)".  None of this attempts is satisfying, for they always use undefined clauses as  'infinite sum',  'formal sum',  'obtained by adding all those terms together',  'if we try to add the terms of...we get...'  'summation'.

The word 'series' is used by mathematicians, yes!  (Although there are complete textbooks on calculus, intentionally totally avoiding this word.)  So readers of Wikipedia should be offered a clear explanation of how how to interprete this word when occurring in a mathematical text. (My personal POV.) --

Invoeren van mijn alternatief, tot aan "Examples", te wijzigen is "Examples of the use of the series representation', of 'series expressions', and of the word 'series' in different situations" .

'Series': adjective and noun

Adjective in combination with: 'expression', 'representation', expansion'. Noun as - a contraction of 'series expression'
-as synonym with 'seqence', - as part of traditional wordings: (absolute) convergent series, convergence test for series, Cauchy product of two series, meaning ............

Ganz klar ?

Jeder Folge ist identisch mit der Partialsummenfolge seiner Differenzenfolge, also:  'Reihe'  und  'Folge'  sind synonym. Deswegen sagt Satz 4 der Definition: "Falls die Folge/Reihe $(s_{n})$ konvergiert, so nennt man die  Grenzwert der Folge/Reihe $(s_{n})$ auch Summe der Folge/Reihe $(s_{n})$" .
Korrekt?   Was ist hier definiert?

Zur Zeit wird eine Alternative diskutiert im 'Talk page' der englische Wikipedia.
--

Das Wort 'Reihe' ist im Mathematik die Nahme für......?

Hej/Hallo. Hier versucht ein Holländer auf Deutsch zu schreiben.
Ich (*1942) bin schon sehr lange interessiert in die Frage um das Unterschied (falls existierend) zwischen 'Folge' und 'Reihe'. Heute entdeckte ich diese Wiki-Seite, wo ich meine Frage sehr ausführlich behandelt sehe. Aber am Ende lese ich doch wieder im Definition (Reihe):
'Reihe' ist die Nahme der Schreibweise (in Symbolform)   $\sum _{k=1}^{\infty }a_{k}$   für (in Textform) "die Partialsummenfolge der Folge $a$ ".
Folglich lese ich im Definition (Grenzwert einer Reihe):
Die Grenzwert einer Schreibweise $\sum _{k=1}^{\infty }a_{k}$, ist der Limes ......
Meine ewige Frage bleibt: Wie kann eine Schreibweise (englisch: a symbolic expression) einen Grenzwert haben? (oder: konvergent sein, oder eine Summe haben, oder konvergieren, oder ...). Das ist doch Unsinn?

Für meine Versuche die Sache auf zu klären, sehe Ganz klar ? oder An attempt to clarify the 'series' mystery
-- Hesselp 12:05, 19. Apr. 2017 (CEST)

Die Definition von 'Reihe' im heutigen Text lautet:

Für eine reelle Folge $(a_{k})_{k\in \mathbb {N} }$ ist die Reihe $\sum _{k=1}^{\infty }a_{k}$ die Folge aller Partialsummen $\left(\sum _{k=1}^{n}a_{k}\right)_{n\in \mathbb {N} }$.

Hier wird (meiner Meinung) nichts anderes gesagt als:

Mit anderen Worten:

$\sum _{k=1}^{\infty }a_{k}$  ist eine (kürzere) Schreibweise für die Partialsummenfolge einer reelle Folge $(a_{k})_{k\in \mathbb {N} }$;
ein solcher Ausdruck - die Sigma-Schreibweise für Folgen - wird Reiheform oder Reihe genannt.

Korte titel: Paraphrasierung der Reihe-Definition.

Verdediging bij skippen: Es soll zumindestens erklärt werden warum die Paraphasierung nicht richtig ist.

Bij reactie "overbodig": oude regel vervángen door de nieuwe. De oude is impliciet en daarom erg onduidelijk.

Stephan Kulla; 21 April

Stephan Kulla   Reaktion zu "sollte anstelle von einer Reihe, von einer Folge sprechen?

Tja, eigentlich: ja!. Oder....obwohl...historisch gesehen sind die Wörte 'Reihe' und 'Folge' sehr oft wie Synonyme gebraucht (Und auch heute noch: sehe z.B. [1] "Folge und Reihe sind also nicht scharf voneinander trennbar. Die Zeitreihen der Wirtschaftswissenschaftler sind eigentlich Folgen."
Und sehe Esperanto:   Rimarko: Ne ekzistas formala diferenco inter la nocioj vico kaj serio. )
Darum kann man auch vorschlagen beide Wörte durcheinander zu benutzen.
Nochmals:
Die 'Folge der Partialsummen einer Folge' ist wieder eine Folge.
Der Limit einer 'Folge der Partialsummen einer Folge' ist ein Zahl.
E s   g i b t   k e i n   m a t h e m a t i s c h e   B e g r i f f   d a z w i s c h e n / d a n e b e n .
Sehr viele Autoren schreiben und sprechen als ob es ein solcher Zwischenbegriff (ich sagte: 'Gespenst') gibt.   Aber wenn man genau analysiert wie das 'Zwischenbegriff' (vielmals 'Reihe' genannt) definiert wird, dann kommt man nicht weiter als: "die Sigma-Schreibweise für Folgen". (Und dazu meistens auch: die "plusses/bullets notation" a1+a2+a3+··· . (Plusse/Punkte-Schreibweise ?))

Reaktion zu: "Ich verstehe das Problem noch nicht."
Das Wort 'Reihe' wird definiert als die Nahme einer symbolisch geschrieben mathematische Ausdruck (die Sigma-Schreibweise). Aber überall im Text wird gesprochen von 'konvergierende Reihe', 'konvergente Reihe', 'Summe der Reihe', 'Partialsummen der Reihe', usw.   Wobei nirgends gesagt wird das der Wikibook-Leser die Schreibweise-Definition vergessen muss ('Reihe' wird niemals in dieser Auffassung gebraucht), und das traditionelle 'Summe der Reihe' muss lesen wie 'Summe der Folge'. Und 'konvergente Reihe' wie 'summierbare Folge'.   Es kann dabei darauf hingewesen werden das Cauchy sehr unvorsichtig 'convergente' wählte als adjective für eine Zahlenfolge mit konvergierende Partialsummen (eine summierbare Zahlenfolge). --

Proposal

This page is about infinite sums. For finite sums, see Summation.

In mathematics (calculus), the word series is primarily used as adjective specifying a certain kind of expressions denoting numbers (or functions).
Symbolic forms like    $a_{1}+a_{2}+a_{3}+\cdots$    and    $\sum a$  or  $\sum _{n=1}^{\infty }a_{n}$   expressing a number as the limit of the partial sums of sequence $a$, are called series expression.  'Series expression' is often shortened to just 'series'.

Secondly, series is used as an adjective in series representation, denoting the kind of representation (of a number or a function) as a limit of the partial sums of a given sequence.

Thirdly, series is used, again as an adjective, in series expansion. Being a special type of series representation (of functions, not numbers).   For instance:
the Maclaurin expansion of a given function  and  the Fourier expansion of a given function  are series expansions.

Finally, (the noun) series can be synonymous with sequence.  Cauchy defined the word series by "an infinite sequence of real numbers".   The use of the word 'series' for 'sequence' has a long tradition, with analogons in other languages, but seems to be considered as somewhat outdated.

The rather widespread idea about the existence of a mathematical notion (a definable mathematical object, called 'series'), 'associated' in some way with a given number sequence, with its partial sums sequence, and with the eventual limit thereof, is false.

The study of the series representation is a major part of mathematical analysis. With this tool, irrationals can be described/defined by means of (the limit of) a relatively easy descriptable sequence of rationals.  
This kind of representation is used in most areas of mathematics, even for studying finite structures (such as in combinatorics), through generating functions. In addition to their ubiquity in mathematics, the series representation is also widely used in other quantitative disciplines such as physics, computer science, statistics and finance.

Situations in which the word  'series'  is used

Definitions, common wordings

Given a infinite sequence $a$  with terms $a_{1},a_{2},a_{3}$ et cetera (or starting with $a_{0}$) for which addition is defined, the sequence
$\quad a_{1},\quad a_{1}+a_{2},\quad a_{1}+a_{2}+a_{3},\ \ ...$     is called  the sequence of partial sums of sequence $a$ .
Alternative notation:     $(a_{1}+\cdots +a_{n})_{n=1,2,\cdots }$  .   Alternative name: the sum sequence of (sequence) $a$ .
Example: The sequence (1, 2, 3, 4, ···)  is the sum sequence of (1, 1, 1, 1, ··· ); being the sum sequence of (1, 0, 0, 0, ··· );   this can be extended in both directions.

A series, short for series expression, is a written expression using mathematical signs, consisting of
- an expression denoting the function that maps a given sequence on the limit of its sum sequence,   combined with
- an expression denoting an infinite sequence (with addition and distance defined).
Examples:     $a_{1}+a_{2}+a_{3}+\cdots$  (plusses-bullets notation),       $\sum _{n=1}^{\infty }a_{n}$  (capital-sigma notation).

Sometimes, the same symbolic forms are used to denote the sum sequence of $a$ , instead of the value of its eventual limit.

A sequence with a converging sum sequence is called summable.  The finite limit is called sum of the sequence.

A valid series expression has a summable sequence as its argument (and denotes a value). Otherwise the expression is void. Traditional wordings are: "convergent/divergent series expression" or "convergent/divergent series".

Convergent / divergent series   The combination convergent series shouldn't be interpreted literally, for an expression itself cannot be convergent or divergent.  By tradition  "Σ $a$ is a convergent series"  as well as  "series Σ $a$ converges"  are used to express that $a$ is summable.   Similarly, "Σ $a$ is a divergent series"  and  "series Σ $a$ diverges"  are used to say that $a$ is not summable.

Convergence test for series   Again, this traditional wording cannot be taken literally because 'series' is the name of an expression of a certain kind, not the name of a mathematical notion. An alternative is: summability test for sequences.

Absolute convergent series   This is the traditional naming for a sequence with summable absolute values of its terms. The alternative  absolute summable sequence  is not in common use.

Series Σ $a$  and  sequence $a$  are interchangeable in traditional clauses like:
- the sum of series Σ $a$,     the terms of series Σ $a$,     the (sequence of) partial sums of series Σ $a$,     the Cauchy product of series Σ $a$ and series Σ $b$
- the series Σ $a$ is geometric, arithmetic, harmonic, alternating, non negative, increasing  (and more).

There is no standard interpretation for the limit of series Σ $a$.

Series representation of numbers and functions

In some contexts the word 'series' shouldn't be seen as referring to a certain type of written symbolic expressions, but as referring to a special type of representation of numbers (and functions). Namely: defining a (irrational) number as the limit of the partial sums of a known infinite sequence of (rational or irrational) numbers.  And in the case of functions: defining a function as the limit of the partial sums of an infinite sequence of functions (which are seen as 'easier' or more elementary in one way or another than the function represented by the limit).
As comparable with the idea of series representation (or: infinite sum representation) can be seen:  the continued fraction representation  and  the infinite product representation (for numbers and functions).

R e d u c t i o n   o f   s u m s   a n d     p r o d u c t s
A sum of two numbers given in series representation,
a product of two numbers given in series representation, and
a product of two numbers, one of them given in series representation,
can be reduced according to:
      $\sum$$_{i}\ a_{i}\ +\,\sum$$_{i}\ b_{i}\ =\,\sum$$_{i}\ (a_{i}+b_{i})$
      $\sum$$_{i}\ a_{i}\ \times \,\sum$$_{i}\ b_{i}\ =\,\sum$$_{i}\ (a_{1}b_{i}+\cdots +a_{i}b_{1})$     (sequence $(|a_{i}|)$ or sequence $(|b_{i}|)$ summable)
      $\sum$$_{i}\ a_{i}\ \times \quad \ c\quad \ \ \,=\,\sum$$_{i}\ (a_{i}\,c)$ .
The same applies for functions instead of numbers.

Series expansion of functions

The name 'series expansion' is used for a special type of series representation of functions. (Not applicable to numbers.)
A series expansion is a series representation of a function, using a sequence of power functions of increasing degree, in one of its variables. Or functions like (for example) $x\rightarrow a_{n}(x-b)^{n},\ \ x\rightarrow a_{n}\sin ^{n}x+b_{n}\cos ^{n}x$.
The labels Maclaurin series, Taylor series, Fourier series shouldn't be seen as denoting expressions but rather representations of the type series expansion. So Maclaurin series should be understood as Maclaurin expansion,  Fourier series as Fourier expansion, et cetera.

Power series

The name power series can occur
- as synonym for Maclaurin expansion,  and
- denoting a series expression which includes an expression for a sequence of power functions with increasing degree.

Cauchy as a source of confusion

Cauchy, in his 'Cours d'Analyse' (1821) made an important, but quite subtile, distinction between the meaning of 'to converge' and 'being convergent':
- A sequence (French: suite) can converge to a limit.
- A sequence with converging partial sums, is called convergent by Cauchy (meaning 'summable')
Moreover, an infinite sequence with real numbers as terms, he called a series (French: série).
This imprudent choise caused permanent confusion around the use of the word 'series'  (e.g. in the German translations of 'Cours d'Analyse' of 1828 and 1885)   until the present day.

Remark on the use of  'series'  and  'convergent / divergent'  in the sections below

Below, the words 'series' and 'convergent / divergent' are not always used conform the preceding descriptions. In such cases the context has to be taken into account to track down the intended meaning.

[More sources on the problem with 'series' in books/publications by: professor H. Von Mangoldt, E.J. Dijksterhuis, H.B.A. Bockwinkel, professor N.G. de Bruijn, professor A.C.M. van Rooij, professor D.A. Quadling, Mike Spivack, H.N. Pot; links have to be added. Several of this sources are written in Dutch.] -- Hesselp (talk) 15:38, 16 April 2017 (UTC)

H o w   t o   r e d u c e   c o n f u s i o n
The best thing to do is:  Stop using the word 'series' at all, and say:
(absolute) summable sequence and summability tests,  in stead of: (absolute) convergent series and convergence tests .
Second best is: inform students and readers of Wikipedia about the historical source of the confusion. Let them understand that the existence of any definable notion 'series' (different from 'sequence') is a wide-spread misconception. And train them to interprete (absolute) convergent series as nothing else as (absolute) summable sequence. -- Hesselp (talk) 13:12, 17 April 2017 (UTC)

Motivation for partly substituting the text of "Series (mathematics)"

The present text strongly suggests that there is only one correct interpretation of what is meant by the word 'series' in mathematical texts. That is that the word 'series' is the name for a certain idea / notion / conception / entity. But what IS "it"?
"It" is NOT a number.
"It" is NOT a sequence (= a mapping on N)
"It" is NOT an expression (for the present text says: "a series is represented by an expression)
"It" is NOT a function.
"It" is 'associated' (what's that?) with a sequence.   "It" is sometimes 'associated' with a value.
"It" has terms and partial sums.
"It" can have a limit, a value, a sum.
"It" can be geometric, arithmetic, harmonic, alterating, converging, diverging, absolute converging, and more.

What's in fact the content of this black "it"-box?   It seems to be empty.
I'm going to replace this unsatisfactory text by an alternative introduction. Chiefly identical with what was shown in this Talk page here, 18 April 2017. The only reaction on it was the remark that "Hesselp doesn't understand what A SERIES IS (in mathematics)".  I agree with that. --

Answer to Wcherowi

@Wcherowi

Neither I nor any other editor is obligated to refute your arguments,

Okay, no one is obligated to write any word or sentence on this Talk page. But when someone makes a revert, I expect a clear motivation on why text B is seen to be of higher value for Wikipedia readers than text A. A motivation, taking into account the arguments that are shown before (that's not the same as 'refuting these arguments', for maybe that could be a difficult task in some cases).

just pointing out that your edits are not supported by citations to reliable secondary sources is sufficient for their removal.

I suppose you mean: text A is "not enough supported by ..." (I'll give a list below). Here the question comes up whether or not text B is more / better supported by this kind of sources. "The sources of this section remain unclear"  I read on top of subsection 'Definition' in (the present) text B. That's in line with the impossibility to find any reference to a source, giving a non-contradictory description of the (supposed) notion named by the word 'series'.

You seem to be under the impression that Wikipedia is an appropriate place to publish your views, but it is not.

But what to do, in case one my 'views' coincide with what I consider as a possibility to improve an existing text?

We have very strong guidelines against what you are attempting (WP:NOR and Wikipedia:SYNTHNOT) and beyond that, Wikipedia is not the place to be righting all the wrongs in the world.

I'm attempting to bring into the article a better description of the (diverse) ways the word 'series' is used by mathematicians. Wikipedia guidelines are against that?
In WP:NOR I found (foot-note 1) that 'language' and 'readable online' are not limiting the required sources (on my list there are some in Dutch).   And in SYNTHNOT, line 5, is said: "After all, Wikipedia does not have firm rules."

If you want Wikipedia to represent your point of view, then get it published in some reliable venue and after it is vetted by the mathematical community we will consider it for inclusion here.

You can see the magazine of the Royal Dutch Mathematical Association as reliable? The article "No one can say what serieses are"; 2008 as representing 'my point of view'? And the review article 2009 as (partial) result of the screening by the mathematical community? (Togethe with an increased use of "summable sequence" in Dutch school-books. And in google-hits.)

None of this, by the way, says anything about the merits of your arguments, some points of which I actually agree with.

It is your profound misunderstanding of what Wikipedia is all about that is making some editors antagonistic in this situation.

"profound misunderstanding"?   It seems that your POV differs from mine, on this point.

Secondary sources supporting Hesselp's edits

- E.J. Dijksterhuis, book review (in Dutch), 1926-27 volume 3, no. 3-4, p.98-101: (paraphrased)  "To consider an infinite series as being an expression, seems to be less desirable."

- H.B.A. Bockwinkel, Integral calculus (in Dutch), 1932:   "The expression   u₁ + u₂ + u₃ + ···   or  Σ₁^∞ u_n   is called a infinite series.  About what an author has in mind with respect to the meaning of this expressions, no information is given."

- P.G.J. Vredenduin, article (in Dutch) 1959 vol. 35, no. 2, p. 57-59:   "In Holland, in lessons on mathematics, normally no clear distinction is made between sequences and serieses."

- P.G.J. Vredenduin, article Sequence and series (in Dutch) 1967 pp.22-23:  "The problem how to define the meaning of the word 'series', is evaded by giving definitions for 'convergent series', 'sum of a convergent series' and 'divergent series',  but not for 'series' alone."

- M. Spivak, Calculus (editions 1967-2006):   "The statement that  {a_n}  is, or is not, summable is conventionally replaced by the statement that the series   Σ_{n =1}^∞ a_n   does, or does not, converge. This terminology is somewhat peculiar, because………."

- N.G. de Bruijn, Printed text (in Dutch) of a series of lectures, 1978,  Language and structure of Mathematics:   "The way language is used with respect to serieses, is traditionally bad."

- H.N. Pot, article What serieses are, you cannot say(in Dutch), 2008

- A.C.M. van Rooij, article, review ofWhat serieses are, you cannot say (in Dutch), 2009:  "Instead of convergent serieses, you will have summable sequences, and everything is okay.  A bonus is that you don't use the word 'convergent' in two different ways."

@D.Lazard. Your 'edit summary' on 25 April 2017 says: "Editor's personal opinion not supported by sources". Without specifying the lines in the reverted text, in which you found a 'personal opinion', and in which more sources are needed according to you. In your remarks on this Talk page, you don't say anything more than that D.Lazard and Wcherowi don't agree with the proposed changes. Nothing on the discussion points on this page, posed on 20:01, 17 April 2017(UTC) and on 22:05, 24 April 2017(UTC). That's not taking part in the discussion as meant in WP:BRD, so your revert was not in accordance with that directive.
One more effort to start discussion.
The present text starts with:   "A series is, informally speaking, the sum of the terms of an infinite sequence."   The terms are numbers, and the sum of numbers is again a number. But: no mathematician uses the word 'series' as a synonyme for 'number'.
Please explain why you prefer this first sentence over the alternative:   "In mathematics (calculus), the word series is primarily used as adjective specifying a certain kind of expressions denoting numbers (or functions)."  (Omit  "as adjective"  if you want.) --

It seems like you have identified a Dutch school of thought on this topic. This would probably be good for a paragraph in the article, but certainly not a rewrite.--Bill Cherowitzo (talk) 05:20, 27 April 2017 (UTC)

@Wcherowi.   Your remark on a 'Dutch school of thought', I cannot see as a way of participating in a discussion on the merits of certain wordings in version A compared with version B.
I'm amazed that an attempt to distinguish different meanings of the word 'series' in the vocabulary of mathematicians (instead of going on attempting to formulate what a series REALLY IS - handed down by God/Allah -), is judged as you do.
You don't give any reason why the fact that most of the cited sources are written in the language where I live, makes their content  c e r t a i n l y  not suited as base for a rewrite of the opening paragraphs (about 1/6 of the article).
Did you notice that all traditional wordings with 'series' are mentioned in the rewritten version? All of them with there meaning(s) carefully (I hope) explained.

I have not seen any reaction on the discussion points, presented at 20:01, 17 April(UTC) and at 22:05 24 April 2017(UTC). I understand that to make a revert by someone who is not taking part in the discussion on the merits of the two versions, is not in accordance with the directive in WP:BRD.   So I feel free to undo such reverts. And to go on trying to reach a version of this article in which the meanings of the word 'series' as used in mathematical texts, are descripted in a clear and unambiguous way. --

Clear as mud ... eh?

About: expressing a number or a function by means of an infinite series. See:
[2]

- The authors of the texts behind the 40 000 google-hits with <summable sequence> and <summable sequences>. --

Talk-page 27 april

@MrOllie.   "Clearly no consensus" ?   That's not very clear at all, for the 'reverters' didn't take part in any discussion on the merits of both versions (apart from "Undocumented POV pushing" and the like).
In more detail:   I extensively mentioned weak points and contradictions in the present text on how the meaning of the word 'series' is described. And showed how (according to me) this can be improved. None of the reverters contributed to discussion on this point. See:

- the draft version of the alternative (Elaborating D.Lazard's...)   15:38, 16 April 2017(UTC)

- the 'some problems left' (1 - 5)   20:01, 17 April 2017(UTC)

- the missing meaning of the "it" in a black box   22:05, 24 April 2017(UTC)

- the choice of the first sentence in the article, answering D.Lazard   23:34, 26 April 2017(UTC) .

The suggestion (Wcherowi) to add the alternative descriptions as a supplement, is an option but maybe not the most desirable.
Concrete arguments contra the present text being shown,  and concrete arguments contra the alternative being absent, I still see the undo of the revert(s) as sufficiently motivated and supported. --

No religion

Mathematics, not religion

The present text presents in the intro plus subsection Definition, four different 'definitions', all of them using the wording:
"a series  IS  ..." .

1. (Intro, sentence 1)   "a series  IS  ... the sum of the terms of ..."
(Being the sum of numbers again a number, the words 'series' and 'number' are synonym.)

2. (Intro, sent.5)   "The series of (associated with) a given sequence a   IS  the expression  a₁+a₂+a₃+··· "
(The word 'series' used as the name of a mapping.)

3. (Definition, sent.1)   "a series  IS  an infinite sum, which is represented by a written symbolic expression of a certain type."
(It isn't clear whether or not the clause after the comma is part of the definition. 'IS' a series still an infinite sum, in situations where it is not represented by an expression of the named form?)

4. (Definition, sent.6)   "series(pl)  ARE  elements of a total algebra of a ring over the monoid of natural numbers over the a commutative ring of the a's "
(The word 'series' as the name for elements of a certain structure; just as the word 'number' is used as the name for elements of another mathematical structure.   To which element in this 'definition' is referred by the a's ? )

In case it is true, that the word 'series' has four different meanings in mathematics (is used in four different ways) the article headed by "Series" should be structured like:
a. The word 'series' is used as name/label for ......... .
b. The word 'series' is also used as name/label for ......... .
c. The word 'series' is used as name/label for .......... as well.
d. Moreover, sometimes the word 'series' is used as name/label for ......... .

The present text directs the reader to believe that there is ONE and only ONE sacred given-by-God-meaning of this word.
That's religion, not mathematics.
Do you think, Wcherowi, the summing up of different meanings is wrong?
Do you think, D.Lazard, the summing up of different meanings is wrong?
Do you think, MrOllie, the summing up of different meanings is wrong?
Do you think, Sławomir Biały, the summing up of different meanings is wrong?

One of the main reasons I see the present text as ready for improvement, I described earlier in

::::::::::this edit in Talk  (to open: press the [show]-button)

The present text strongly suggests that there is only one correct interpretation of what is meant by the word 'series' in mathematical texts. That is that the word 'series' is the name for a certain idea / notion / conception / entity. But what IS "it"?

"It" is NOT a number.
"It" is NOT a sequence (a mapping on N)
"It" is NOT an expression (for the present text says: "a series is represented by an expression)
"It" is NOT a function.
"It" is 'associated' (what's that?) with a sequence.   "It" is sometimes 'associated' with a value.
"It" has terms and partial sums.
"It" can have a limit, a value, a sum.
"It" can be geometric, arithmetic, harmonic, alterating, converging, diverging, absolute converging, and more.

What's in fact the content of this black "it"-box?   It seems to be empty.
I'm going to replace this unsatisfactory text by an alternative introduction. Chiefly identical with what was shown in this Talk page here, 18 April 2017. The only reaction on it was the remark that "Hesselp doesn't understand what A SERIES IS (in mathematics)".  I agree with that. --Hesselp (talk) 22:05, 24 April 2017 (UTC)

--

Critical remarks on the first twelve sentences of edit 30 April 2017, 14:59

1. (Sent.1)   "a series  IS  ... the sum of the terms of ..."
Being the sum of numbers again a number, the words 'series' and 'number' are used as synonym. A few lines later it is said that this is not intended.

2. (Sent.2)   "a series continues indefinitely"
What is meant by:   an indefinitely continuing 'sum of the terms of something' ?

3. (Sent.4)   "the value of a series"
What is meant by:   the value of a sum (a number) ?

4. (Sent.4)   "evaluation of a limit of something"
What's meant with this?
Is it true that a series doesn't have a value, without that limit being 'evaluated' ?
Is it always possible to 'evaluate' the limit of a sequence of terms ?

5. (Sent.5)   "the expression obtained by adding all those (an infinite number of) terms together"
A (symbolic, written) expression can be obtained by writing down some symbols using a pen or pencil (or using the keys of a keyboard). The task of adding an infinite number of terms is not feasible, so never any expression will be obtained.

6. (Sent.6)   "obtained by placing the terms $a_{n}$ side-by-side with pluses in between them.
This 'placing' sounds much better feasible. I miss the three centered dots ('bullets') at the right end.

7. (Sent.6)   "infinite expression"
I see 'series' and 'infinite sum' used as synonyms for 'infinite expression'. But what notion / mathematical object is denoted by this labels ? It must be a notion 'not being a part of the conventional foundations of mathematics'.  How many readers of this article are acquainted with this notion already by themselves?

8. (Sent.7)   "The infinite expression can be denoted ..." Such expressions mostly denote a number, a function or a sequence. But an expression denoting a expression sound very strange.

9. (Sent.9)   "two series of the same type"
I cannot find where is explained what is meant by:  'the type of that mysterious notion called series '.

10. (Sent. 8, 9, 10, 11, 12)
Is the (intended) information communicated by this five sentences really of enough importance to be incorporated in the 'introduction' ?

11. (First line after 'Definition') The twofold description of the meaning of the word 'series' (as sum, and as expression) causes - unnecessary? - complexity.
--

To Slawomir

@Slawomir, and maybe other readers of this Talk page.
You write that you don't want to continue discussion; it's your choice. This doesn't prevent me from writing down my comments on what you put forward.

1)   About the 'mysterious' status of the notion/concept named 'series'.
I used the word 'mysterious' to refer in a short way to the   "it" is NOT a ....-list.  It was and is not meant as sarcastic.
On 30 April, 14:30 and 21:46 you're argumenting your view that  "there IS a (one) concept of series".   My hesitations to agree with you on this point, have to do with your formulations (wordings) like:
- it is often useful to build a model of series ...       - This is an "interpretation" of "series" ...       - Series are not formally axiomatized ...    - which includes the concept of mathematical series     - But series do exist ... to build a model of them.
Here you are suggesting every time that you have an a priori believe in the existence of a notion named 'series'.
There are believers, and there are non-believers.

2)   About "an expression denoting an expression".   To me this sounds still as strange as before.
You attempt to explain this by: "The sigma notation refers to the infinite expression".  But isn't it universally agreed that a sigma expression denotes (= refers to) a number (more general: a function) or a sequence?  Not an expression.

3)   About:  "The basic definition is ... a bunch of terms with plus signs placed between".
I see this as being very close to sentence 2-3 in my edit 21:24 28 April 2017(UTC):
        Symbolic forms like    $a_{1}+a_{2}+a_{3}+\cdots$    and    $\sum a$  or  $\sum _{n=1}^{\infty }a_{n}$   expressing a number as the limit of the
        partial sums of sequence $a$, are called series expression.  'Series expression' is often shortened to just 'series'.
I use the short notations $a$ for a mapping on N (a sequence) and $\sum a$ as alternative for $\sum _{n=1}^{\infty }a_{n}$ (avoiding problems with the first index). I know that this is not usual, so if this is seen as not desirable I don't persist.
My choice of wordings at some places has to do with my view on expressions in general: verbal expressions versus written expressions,  and written expressions using text versus written expressions using mathematical symbols.

4)   About:  "To be very precise, we should say that the expression "1+1" evaluates to the number "2" .
I think it's better to say:
the expressions "1+1" and "2" are equivalent (equi-valent = same value); or
the expression "1+1" can be rewritten as  "2" ; or
the expression "1+1" can be reduced to  "2" ; op
the standard form for the value of expression "1+1" is  "2" .
The meaning of "the evaluation of an expression" is not clear (to me). The expression  "e+π"  denotes (refers to) a certain (irrational) number. So the expression has a value. But the expression does not 'evaluate to a number' . --

To evaluate a given expression means ... ?

@Slawomir.   Never in my life I've denied that mathematical expressions are totally different from numbers. You must have misunderstood me somewhere, I cannot trace back where this could have happened.
I agree with you on everything you wrote in the first 7 sentences in 12:46, 2 May 2017(UTC)   (Until "The sigma notation for a series..."). About your sentences 8, 9, 10  I'm not sure. Maybe things become more clear from your judgment of the following statements a - h (true or false):

a)   the expression   e+π   evaluates to (= has as its value) the number   e+π

b)   the expression   1+1   evaluates to the number   1+1

c)   the expression   1+1   evaluates to the number   2

d)   the sigma expression   Σ_{i =1}^∞ a_i   evaluates to the infinite expression   a₁+a₂+a₃+···

e) Provided that   lim_n→∞ (a₁+ ··· +a_n)  exists,
   in other words   lim_n→∞ (a₁+ ··· +a_n)  is a valid expression,
   in other words   sequence (a_n)  is summable,
      the infinite expression   a₁+a₂+a₃+··· (number-interpretation)   evaluates to the number   lim_n→∞ (a₁+ ··· +a_n)

f)   the infinite expression   a₁+a₂+a₃+··· (sequence-interpretation)   evaluates to the sequence   (a₁+ ··· +a_n)_n≥1

g) Being p₁, p₂, p₃, ··· successive primes,
      the infinite expression   p₁^-3+ p₂^-3 + p₃^-3+ ···   evaluates to the number   p₁^-3+ p₂^-3 + p₃^-3+ ···

h)   the infinite expression   9^{− 9^1}+ 9^{− 9^2}+ 9^{− 9^3}+ ···   evaluates to the number   Σ_{i =1}^∞ 9^{− 9^í}

According to me this is a quite peculiar way to use the verb 'to evaluate' (in the intro of the present text: "A series is thus evaluated by examining ...."); you can show sources?   I only saw it, meaning: given an expression (denoting a number), find the decimal representation of its value, exact or approximated. --

An 'infinite expression' is an expression with infinite physical dimensions, or ... ?

The intro of the present text explains the meaning of "series" using:
The series of an given infinite sequence is the infinite expression that is obtained by placing terms side-by-side with pluses in between.
By 'infinite expression' is not meant an expression with infinite physical dimensions. Nor an expression of the type "1/0".
The Wikipedia article says: "an expression in which some operators take an infinite number of arguments".   That's sufficiently clear to most of our readers?  I doubt
Moreover, that article has: "Examples of well-defined infinite expressions include infinite sums, whether expressed using summation notation or as an infinite series, ....". With a circulating reasoning, because 'infinite sum' is linked to the article named ....'Series (mathematics)'.   --

To D.Lazard

@D.Lazard.   Your post dated 15:14, 30 April 2017(UTC) starts referring  that this article is about the mathematical concept named 'series'.  Okay.
What I'm trying is:  to improve the description in the article of what is considered by mathematicians as the content of this concept ('mathematical object', as you say).   That's legal?
You wrote:  "a rigorous definition is too technical for being understood by beginners".   In my view a considerable reduction of this difficulties is furnished by skipping a number of generalizations of the original concept. By restricting (in the first part of the article) to serieses associated with real (or complex) sequences and real (or complex) functions. And with the plus sign only denoting the traditional addition.   You agree with this restriction?

I don't know whether this will be enough to make it possible to present a 'rigorous' definition of the (restricted) concept.  If not, be open/honest to the reader: say that a complete description is not presented here, and show references to other sources within or without Wikipedia.
And tell the reader that they can 'drive the car'  by reading  "series a₁+a₂+a₃+ ··· is (not) convergent"
as   "sequence  (a₁+···+a_n) _{n ≥1}  converges" .
In words (suited to verbal communication):   "sequence a1, a2, a3, et cetera  is (not) summable" .   (Without the need to understand fully the deep rooted concept 'series'.)   Any objections? --

The "it"-is-NOT-list;   negative statements on "series"

In the discussion on the concept/object/idea/entity (mathematical or philosophical) named "series", a number of negative statements are made on this Talk page, since April 10, 2017.  Two new ones are found in a post by Sławomir Biały, dated 21:57, 2 May 2017(UTC):     - "it" is NOT a numeral     - "it" is NOT an expression that denotes a number.

Press the [show]-button for the complete list

"It" is NOT a number.

"It" is NOT a sequence (a mapping on N)
"It" is NOT an expression
"It" is NOT a function
"It" is NOT a part of Zermelo-Fraenkel set theory
"It" is NOT a part of the conventional foundations of mathematics. "It" is NOT a numeral
"It" is NOT an expression that denotes a number "It" is 'associated' (what's that?) with a sequence.
"It" has terms and partial sums.
"It" can have a limit, a value, a sum.
"It" is represented by an expression
"It" is sometimes 'associated' with a value.
"It" can be geometric, arithmetic, harmonic, alterating, converging, diverging, absolute converging, and more.

--

Bovenstaande is (nog) niet geplaatst!

On edit 15:34, 4 May 2017. What is meant: a series is a description of the operation: adding one-by-one infinitely many quantities
or
a series is the operation: adding one-by-one infinitely many terms ?
What a reader should think of: an operation that cannot be carried on (not 'effectively') ? I'm curious to see how you define (based on reliable sources): "a convergent infinite adding operation",   "a alternating infinite adding operation"   "a geometric infinite adding operation"   "a Fourier infinite adding operation"   "the Cauchy product of two infinite adding operations"   "a power infinite adding operation"   and much more.
Please present a mature proposal for the intro-plus-definition part of the article. Here on Talk page, so not unnecessary disturbing our Wiki-readers . --

@D.Lazard.   Some more remarks.
A.   Footnote 5 in the current version of the article mentions Michael Spivak's book "Calculus" (1st edition 1967, latest(?) 2008). His chapter INFINITE SERIES starts with a box with:
        A sequence is summable if the sequence of its partial sums converges.
        In this case the limit of its partial sums is called the sum of the sequence.
Isn't this extremely close to the wording:
        A sequence with a converging sum sequence (= sequence of partial sums) is called summable.
        The finite limit is called sum of the sequence.
as used in this edit ?   If you know a more preferable alternative for the word 'summable', please show it.

B.   Your view on the 'mathematical object series' , I understand as being:   the operation (evaluation, calculation) producing (if possible) an expression denoting the decimal representation of the sum of a given sequence.
I'll incorporate this view in the text I plan to edit instead of the current one (recently judged as "too technical", "biased", "worth cleaning up", "rather of a mess").

C.   In this post you wrote:
      "It appears that this concept is not a simple one, as it involves the concept of infinity, which was not well
      understood nor well accepted before the end of the 19th century (this make your citation of Cauchy
      irrelevant for discussing the modern terminology; note that he avoided carefully to talk about infinity)."
I don't see your point with  'avoided carefully'.
For in Cauchy's "Cours d'Analyse" (1821) I read on page 4:   "Lorsque ...s'approchent indéfiniment ...est appellée la limite de ... ". (As...approaches infinitely ... is called the limit of ...).   And on the famous/notorious page 123:  "...une suite indéfinie...", "la somme s'approche indéfiniment...d'une certaine limite s", "n croît indéfiniment"   ("an infinite sequence", "the sum approaches indefinitely some limit s", "n increases indefinitely).
I don't see a substantial difference with the 'modern' view. Please elucidate why citing Cauchy as I did, is irrelevant? --

Attempts to start discussion

Motivation for reposting an alternative intro+definition text,  6 May 2017

In Talk page, no user took part in discussion on the merits of the content of this text. So  'no consensus'  cannot be a valid reason to revert.

From the 'edit summary' 13:22, 5 May 2017:   "...most editors have already given up trying to communicate with you" .
That  'trying to communicate'  refers to reactions with no more relation to the content of the proposed text-section, than in phrases of the type:

- don't agree with proposed changes   - undocumented POV-pushing   - Hesslp doesn't understand what a series is   - this talk page is not for discussing personal opinions about the practice of mathematicians   - this is not mathematics, it is philosophy   - you have clearly a misconception of what is mathematics   - for being clearer, every line of Hesselp's post is either wrong, or does not belong to this talk page or both   - I reiterate my objection .

Attempts made to start discussion, in this list:

Some problems left (1 - 5)

Answer to D.Lazard: Thank you for contributing to the search for the best way to describe what is meant with the word 'series' in texts on mathematics(calculus). I saw some points in your rewriting of the Definition section which I can see as improvements. But there are some problems left:
1.   Rewording the first sentence more close to the usual way as definition of 'infinite series / series',  I get:
An infinite sum is called series or infinite series if represented by an expression of the form: $a_{0}+a_{1}+a_{2}+\cdots ,$ . . .
This paraphrasing is correct?
Please add an explanation of what you mean by 'infinite sum'.  And tell how a blind person can decide whether or not he is allowed to say 'series' to such an infinite sum, as he cannot see the form of the representation.
2.   In the third sentence 'summation notation' is introduced, showing a 'capital-sigma' form, followed by an equal sign and a 'plusses-bullets' form. Why two different forms to illustrate the 'summation notation'?
3.   Please explain what you mean with 'formal sum' (fourth sentence). See this discussion. And the same question for 'summation' at the end of that sentence.
4.   Your seventh sentence end with "...the convergence of a series". Do you really mean to define "the convergence of an expression(of a certain type)" ?
5.   Finally, I'ld like to see an explanation of the clause "the expression obtained by adding all those [an infinite number of] terms together" (fifth sentence in the intro). I don't see how the activity of 'adding' (of infinite many terms!) can have an 'expression' as result. -- Hesselp (talk) 20:01, 17 April 2017 (UTC)

The "it"-is-NOT-list;   negative statements on  "series"

The present text strongly suggests that there is only one correct interpretation of what is meant by the word 'series' in mathematical texts. That is that the word 'series' is the name for a certain idea / notion / conception / entity. But what IS "it"?

"It" is NOT a number.
"It" is NOT a sequence ( a mapping on N)
"It" is NOT an expression "It" is NOT a function.
"It" is NOT a part of Zermelo-Fraenkel set theory "It" is NOT an expression that denotes a number "It" is NOT a numeral "It" is represented by an expression
"It" is 'associated' (what's that?) with a sequence.
"It" is sometimes 'associated' with a value.
"It" has terms and partial sums.
"It" can have a limit, a value, a sum.
"It" can be geometric, arithmetic, harmonic, alterating, converging, diverging, absolute converging, and more.

What's in fact the content of this black "it"-box? --Hesselp (talk) 22:05, 24 April 2017 (UTC)

The choice of the first sentence in the article, answering D.Lazard

@D.Lazard. Your 'edit summary' on 25 April 2017 says: "Editor's personal opinion not supported by sources". Without specifying the lines in the reverted text, in which you found a 'personal opinion', and in which more sources are needed according to you. In your remarks on this Talk page, you don't say anything more than that D.Lazard and Wcherowi don't agree with the proposed changes. Nothing on the discussion points on this page, posed on 20:01, 17 April 2017(UTC) and on 22:05, 24 April 2017(UTC). That's not taking part in the discussion as meant in WP:BRD, so your revert was not in accordance with that directive.
One more effort to start discussion.
The present text starts with:   "A series is, informally speaking, the sum of the terms of an infinite sequence."   The terms are numbers, and the sum of numbers is again a number. But: no mathematician uses the word 'series' as a synonyme for 'number'.
Please explain why you prefer this first sentence over the alternative:   "In mathematics (calculus), the word series is primarily used as adjective specifying a certain kind of expressions denoting numbers (or functions)."  (Omit  "as adjective"  if you want.) --Hesselp (talk) 23:34, 26 April 2017 (UTC)

Don't mix four different definition of  "series"

The present text presents in the intro plus subsection Definition, four different 'definitions', all of them using the wording:
"a series  IS  ..." .

1. (Intro, sentence 1)   "a series  IS  ... the sum of the terms of ..."
(Being the sum of numbers again a number, the words 'series' and 'number' are used as synonym.)

2. (Intro, sent.5)   "The series of (associated with) a given sequence a   IS  the expression  a₁+a₂+a₃+··· "
(The word 'series' used as the name of a mapping.)

3. (Definition, sent.1)   "a series  IS  an infinite sum, which is represented by a written symbolic expression of a certain type."
(It isn't clear whether or not the clause after the comma is part of the definition. 'IS' a series still an infinite sum, in situations where it is not represented by an expression of the intended form?)

4. (Definition, sent.6)   "series(pl)  ARE  elements of a total algebra of a ring over the monoid of natural numbers over the a commutative ring of the a's "
(The word 'series' as the name for elements of a certain structure; just as the word 'number' is used as the name for elements of another mathematical structure.   To which element in this 'definition' is referred by "the a's" ? )

In case it is accepted that the word 'series' has four different meanings in mathematics (is used in four different ways) the first part of the article headed by "Series" should be structured like:
a. The word 'series' is used as name/label for ......... .
b. The word 'series' is also used as name/label for ......... .
c. The word 'series' is used as name/label for .......... as well.
d. Moreover, sometimes the word 'series' is used as name/label for ......... .

The present text directs the reader to believe that there is ONE and only ONE sacred given-by-God-meaning of this word.
That's religion, not mathematics.
Do you think, Wcherowi, the summing up of different meanings is wrong?
Do you think, D.Lazard, the summing up of different meanings is wrong?
Do you think, MrOllie, the summing up of different meanings is wrong?
Do you think, Sławomir Biały, the summing up of different meanings is wrong?

-- Hesselp (talk) 13:31, 30 April 2017 (UTC)

Critical remarks on the first twelve sentences of edit 30 April 2017, 14:59.   Nrs. 1 - 11

1. (Sent.1)   "a series  IS  ... the sum of the terms of ..."

Being the sum of numbers again a number, the words 'series' and 'number' are used as synonym. A few lines later it is said that this is not intended.

2. (Sent.2)   "a series continues indefinitely"
What is meant by:   an indefinitely continuing 'sum of the terms of something' ?

3. (Sent.4)   "the value of a series"
What is meant by:   the value of a sum (a number) ?

4. (Sent.4)   "evaluation of a limit of something"
What's meant with this?
Is it true that a series doesn't have a value, without that limit being 'evaluated' ?
Is it always possible to 'evaluate' the limit of a sequence of terms ?

5. (Sent.5)   "the expression obtained by adding all those (an infinite number of) terms together"
A (symbolic, written) expression can be obtained by writing down some symbols using a pen or pencil (or using the keys of a keyboard). The task of adding an infinite number of terms is not feasible, so never any expression will be obtained.

6. (Sent.6)   "obtained by placing the terms $a_{n}$ side-by-side with pluses in between them.
This 'placing' sounds much better feasible. I miss the three centered dots ('bullets') at the right end.

7. (Sent.6)   "infinite expression"
I see 'series' and 'infinite sum' used as synonyms for 'infinite expression'. But what notion / mathematical object is denoted by this labels ? It must be a notion 'not being a part of the conventional foundations of mathematics'.  How many readers of this article are acquainted with this notion already by themselves?

8. (Sent.7)   "The infinite expression can be denoted ..." Such expressions mostly denote a number, a function or a sequence. But an expression denoting a expression sound very strange.

9. (Sent.9)   "two series of the same type"
I cannot find where is explained what is meant by:  'the type of that mysterious notion called series '.

10. (Sent. 8, 9, 10, 11, 12)
Is the (intended) information communicated by this five sentences really of enough importance to be incorporated in the 'introduction' ?

11. (First line after 'Definition') The twofold description of the meaning of the word 'series' (as sum, and as expression) causes - unnecessary? - complexity.
--Hesselp (talk) 23:13, 30 April 2017 (UTC)

"A bunch of terms with plus signs placed between";  with three centered dots it's perfect

1)   About the 'mysterious' status of the notion/concept named 'series'.
I used the word 'mysterious' to refer in a short way to the   "it" is NOT a ....-list.  It was and is not meant as sarcastic.
On 30 April, 14:30 and 21:46 you're argumenting your view that  "there IS a (one) concept of series".   My hesitations to agree with you on this point, have to do with your formulations (wordings) like:
- it is often useful to build a model of series ...       - This is an "interpretation" of "series" ...       - Series are not formally axiomatized ...    - which includes the concept of mathematical series     - But series do exist ... to build a model of them.
Here you are suggesting every time that you have an a priori believe in the existence of a notion named 'series'.
There are believers, and there are non-believers.

2)   About "an expression denoting an expression".   To me this sounds still as strange as before.
You attempt to explain this by: "The sigma notation refers to the infinite expression".  But isn't it universally agreed that a sigma expression - in case it is not meaningless/void - denotes / refers to  a number (more general: a function) or a sequence?  Not an expression.

3)   About:  "The basic definition is ... a bunch of terms with plus signs placed between".
I see this as being very close to sentence 2-3 in my edit dated 21:24 28 April 2017:
        Symbolic forms like    $a_{1}+a_{2}+a_{3}+\cdots$    and    $\sum a$  or  $\sum _{n1}^{\infty }a_{n}$   expressing a number as the limit of the
        partial sums of sequence $a$, are called series expression.  'Series expression' is often shortened to just 'series'.
I use the short notations $a$ for a mapping on N (a sequence) and $\sum a$ as alternative for $\sum _{n1}^{\infty }a_{n}$ (avoiding problems with the first index). I know that this is not usual, so if this is seen as not desirable I don't persist.
My choice of wordings at some places has to do with my view on expressions in general: verbal expressions versus written expressions,  and written expressions using text versus written expressions using mathematical symbols.

4)   About:  "To be very precise, we should say that the expression "1+1" evaluates to the number "2" .
I think it's better to say:
the expressions "1+1" and "2" are equivalent (equi-valent ≈ same value); or
the expression "1+1" can be rewritten as  "2" ; or
the expression "1+1" can be reduced to  "2" ; or
the standard form for the value of expression "1+1" is  "2" .
The meaning of "the evaluation of an expression" is not clear (to me). The expression  "e+π"  denotes (refers to) a certain (irrational) number. So the expression has a value. But the expression does not  'evaluate to a number' . --Hesselp (talk) 21:22, 1 May 2017 (UTC)

Statements a-h,  true or false ?

@Sławomir Biały.   Never in my life I've denied that mathematical expressions are totally different from numbers. You must have misunderstood me somewhere, I cannot trace back where this could have happened.

I agree with you on everything you wrote in the first seven sentences in 12:46, 2 May 2017(UTC)   (Until "The sigma notation for a series..."). About your sentences 8, 9, 10  I'm not sure. Maybe things become more clear from your judgment of the following statements a - h (true or false, or ...):

a)   the expression   e+π   evaluates to (has as its value) the number   e+π

b)   the expression   1+1   evaluates to the number   1+1

c)   the expression   1+1   evaluates to the number   2

d)   the sigma expression   Σ_{i 1}^∞ a_i   evaluates to the infinite expression   a₁+a₂+a₃+···

e) Provided that   lim_n→∞ (a₁+ ··· +a_n)  exists,
   in other words   lim_n→∞ (a₁+ ··· +a_n)  is a valid expression,
   in other words   sequence (a_n)  is summable,
      the infinite expression   a₁+a₂+a₃+··· (number-interpretation)   evaluates to the number   lim_n→∞ (a₁+ ··· +a_n)

f)   the infinite expression   a₁+a₂+a₃+··· (sequence-interpretation)   evaluates to the sequence   (a₁+ ··· +a_n)_n≥1

g) Being p₁, p₂, p₃, ··· successive primes,
      the infinite expression   p₁^-3+ p₂^-3 + p₃^-3+ ···   evaluates to the number   p₁^-3+ p₂^-3 + p₃^-3+ ···

h)   the infinite expression   9^{− 9^1}+ 9^{− 9^2}+ 9^{− 9^3}+ ···   evaluates to the number   Σ_{i 1}^∞ 9^{− 9^í}

According to me this is a quite peculiar way to use the verb 'to evaluate' (in the intro of the present text: "A series is thus evaluated by examining ...."); you can show sources?   I only saw it, meaning: given an expression (denoting a number), find the decimal representation of its value, exact or approximated. --Hesselp (talk) 21:37, 2 May 2017 (UTC)

Unfortunately you made no judgments (true, false, ...) at the statements a - h.

Slawomir Bialy, Unfortunately you made no judgments (true, false, ...) at the statements a - h. That makes it difficult, if not impossible, for me and others, to understand the ratio of your critisims.
You prefer 'numeral' over the longer 'an expression denoting a number'.  Okay, perfect.
But I don't grasp why you declare:   the expression "1+1" is NOT a numeral. (neither is the expression "1+1")
For in your post dated 12:46, 2 May 2017(UTC), you started with:
- the expression "1+1" evaluates to/has value the number  "2" .
Is there anyone who can explain why
- the expression  1+1  denotes the number  2,   and
- the expression  e+π  denotes the number  e+π ,
should not be correct as well? --Hesselp (talk) 20:00, 3 May 2017 (UTC)

3 questions:   That's legal?   Agree with restriction?   Any objections?

@D.Lazard.   Your post dated 15:14, 30 April 2017, starts referring  that this article is about the mathematical concept named 'series'.  Okay.
What I'm trying is:  to improve the description in the article of what is considered by mathematicians as the content of this concept ('mathematical object', as you say).   That's legal?
You wrote:  "a rigorous definition is too technical for being understood by beginners".   In my view a considerable reduction of this difficulties is furnished by skipping a number of generalizations of the original concept. By restricting (in the first part of the article) to serieses associated with real (or complex) sequences and real (or complex) functions. And with the plus sign only denoting the traditional addition.   You agree with this restriction?

I don't know whether this will be enough to make it possible to present a 'rigorous' definition of the (restricted) concept.  If not, be open/honest to the reader: say that a complete description is not presented here, and show references to other sources within or without Wikipedia.
And tell the reader that they can 'drive the car'  by reading  "series a₁+a₂+a₃+ ··· is (not) convergent"
as   "sequence  (a₁+···+a_n) _{n ≥1}  converges" .
In words (suited to verbal communication):   "sequence a1, a2, a3, et cetera  is (not) summable" .   (Without the need to understand fully the deep rooted concept 'series'.)   Any objections? --Hesselp (talk) 10:36, 4 May 2017 (UTC)

Is a series a description of an operation, or is it the operation itself ?

@D.Lazard.   What is meant:
a series is  a description  of the operation: adding one-by-one infinitely many quantities (line 1)
or
a series is  the operation : adding one-by-one infinitely many terms (line 16) ?
What a reader should think of: an operation that cannot be carried on (not 'effectively') ?
I'm curious to see how you define (based on reliable sources): "a convergent infinite adding operation",   "a alternating infinite adding operation"   "a geometric infinite adding operation"   "a Fourier infinite adding operation"   "the Cauchy product of two infinite adding operations"   "a power infinite adding operation"   and much more.
Please present a mature proposal for the intro-plus-definition part of the article. Here on Talk page, so not unnecessary disturbing our Wiki-readers . --Hesselp (talk) 17:30, 4 May 2017 (UTC)

What phrases used by me you see as strange and incorrect?

D.Lazard, you don't answer my first question: is a series a 'description' or an 'operation'?
On your statement: "the square root operation is in many cases an operation that cannot be done effectively" (I'm inclined to say for short: an impossible operation, a void operation)  I plan to come back later. You are right of course when I interprete "operation", just as "calculation" and "evaluation", as: rewriting a number (or a function) given in the limit-of-the-sum-sequence-of-a-given-sequence-representation,  into the well known decimal representation.
About your last remark: please be concrete, and tell what formulations you see as 'never used', and what phrases used by me you see as strange and incorrect.   What's wrong and what's incorrect with:

"sequence  (a₁+···+a_n) _{n ≥1}  converges"     or    "sequence  a₁,   a₁+a₂,   a₁+a₂+a₃,  ···  converges" ,

and (better suited to verbal communication):   "sequence a1, a2, a3, et cetera  is (not) summable" ?

You've seen the number of hits by Google for <summable sequence> and <summable sequences> ? Quite remarkable is the much lower number of hits for <suite sommable> in French. -- Hesselp (talk) 22:57, 4 May 2017 (UTC)

Some more remarks / questions: A, B, C

@D.Lazard.
A.   Footnote 5 in the current version of the article mentions Michael Spivak's book "Calculus" (1st edition 1967, latest(?) 2008). His chapter INFINITE SERIES starts with a box with:
        A sequence is summable if the sequence of its partial sums converges.
        In this case the limit of its partial sums is called the sum of the sequence.
Isn't this extremely close to the wording:
        A sequence with a converging sum sequence (sequence of partial sums) is called summable.
        The finite limit is called sum of the sequence.
as used in the alternative edit ?   If you know a more preferable alternative for the word 'summable', please show it.

B.   Your view on the 'mathematical object series' , I understand as being:  the operation (evaluation, calculation) producing (if possible) an expression denoting the decimal representation of the sum of a given sequence.
I'll incorporate this view in the text I plan to edit instead of the current one (recently judged as "too technical", "biased", "worth cleaning up", "rather of a mess").

C.   In your post of 15:14, 30 April 2017 you wrote:
      "It appears that this concept is not a simple one, as it involves the concept of infinity, which was not well
      understood nor well accepted before the end of the 19th century (this make your citation of Cauchy
      irrelevant for discussing the modern terminology; note that he avoided carefully to talk about infinity)."
I don't see your point with  'avoided carefully'.
For in Cauchy's "Cours d'Analyse" (1821) I read on page 4:   "Lorsque ...s'approchent indéfiniment ...est appellée la limite de ... ". (As...approaches infinitely ... is called the limit of ...).   And on the famous/notorious page 123:  "...une suite indéfinie...", "la somme s'approche indéfiniment...d'une certaine limite s", "n croît indéfiniment"   ("an infinite sequence", "the sum approaches indefinitely some limit s", "n increases indefinitely).
I don't see a substantial difference with the 'modern' view. Please elucidate why citing Cauchy as I did, is irrelevant? -- Hesselp (talk) 11:38, 5 May 2017 (UTC)

--

Administrators

User:Hesselp

There is a situation with Hesselp (talk · contribs · deleted contribs · logs · filter log · block user · block log) on the page Series (mathematics) and the talk page Talk:Series (mathematics). He has been edit-warring to include his rewrite of the article [3], [4], [5], [6], [7], [8]. Although not at the moment above 3RR, the above is clear indication of edit warring, being reverted by four different editors. He was warned against edit warring, yet persists. Other editors have attempted to engage him at Talk:Series (mathematics), but attempts to resolve the dispute amicably are met with walls of antagonistic rambling text: [9], [10], [11], [12], [13], among others. We have given up on trying to interact with this user, in the spirit of WP:DENY (the above posts strongly suggest trolling). But I believe the time has come for this disruption to be put to an end administratively. (Pinging other involved editors: @Hesselp:, @D.Lazard:, @MrOllie:, @Wcherowi:.) Sławomir Biały (talk) 11:58, 6 May 2017 (UTC)

Reaction by Hesselp.   I haven't done anything else than concentrate on the best way - at the level of mainstream Wikipedia readers - to describe the meanings (plural) of the technical/mathematical term "series" in mathematical texts. A main point is that the meaning of "convergent series" can be explained easily by interpreting this words as "summable sequence". This is not at all new, see the number of google-hits on "summable sequence" and "summable sequences". The same point is shown in Calculus by M. Spivak (editions 1968-2008).   To which extend it is reasonable to characterize my posts on Talk page as "rambling antagonistic text", I leave to decide by other judges.
@Slawomir Bialy: my edit is not a "rewrite of the article", it can be seen as a rewrite of 1/6 of the article.
@MrOllie: Yes, I tried about the same on Dutch Wikipedia, with partial success.
@Wcherowi: - (on your newest 'edit summary') Using  'no consensus'  without ANY discussion on the merits of the content of a text/edit, is misusing this word.  - 40 000 hits on 'summable sequence(s)' does NOT point to an "extreme position".   - Tell me at least, which aspect(s) in the edit you see as 'extreme', it's certainly by far not the complete text. --

Toegevoegd vraag aan L3X1 (?)

"Mathematicians agree on the concept of a series".  Is this true?

D.Lazard writes (15:14, 30 April 2017):   "Presently, mathematicians agree on the concept of a series, but as usual for concepts that have many applications, the formal rigorous definition is too technical for being understood by beginners, .....".
This 'agree on' seems to be not in accordance with the ongoing rewriting of the Definition section in the article. Not with the absence of a decisive unambiguous source. And not with the result of a survey, made around 2008. About eighty books on calculus were inspected, the results are shown below (press [show]). The original language was not always English; capital-sigma forms were seen as not different from  a₁ + a₂ + a₃ + ··· .

                  32 definitions

1. An (infinite) series is   an expression of the form   a₁ + a₂ + a₃ + ···

Bowman, Britton/Kriegh/Rutland, Edwards/Penny, Open University-UK, Small/Hosack

2. An (infinite) series is   an expression that can be written in the form   a₁ + a₂ + a₃ + ···
Anton/Herr, Anton, Anton/Bivens/Davis

3. An (infinite) series is   a formal sum of infinitely many terms.
R A Adams

4. An (infinite) series is   a formal infinite sum.
Ahlfors

5. The formal expression   a₁ + a₂ + a₃ + ···   is called an (infinite) series.
Matthews/Howell, Sherwood/Taylor

6. An (infinite) series is   an indicated sum of the form   a₁ + a₂ + a₃ + ···
Kaplan

7. An (infinite) series is   a sequence   a₁,   a₁ + a₂,   a₁ + a₂ + a₃, ···
Hurley

8. An (infinite) series is   a sequence whose terms are to be added up.
Marsden/Weinstein

9. An (infinite) series is   the indicated sum of the terms of a sequence.
Daintith/Nelson, Kells, Weber

10. An (infinite) series is   the sum of the terms of a sequence.
Wikipedia-Spanish

11. An (infinite) series is   the sum of a sequence of terms.
Borowski/Borwein

12. An (infinite) series is   the sum of an infinite number of terms.
Lyusternik/Yanpol'skii

13. An (infinite) series is   a sum of a countable number of terms.
Borden

14. An (infinite) series is   an infinite addition of numbers.
Goldstein/D C Lay/Schneider(/Asmar)

15. An (infinite) series is   an infinite sum: a mathematical proces which calls for an infinite number of additions.
Davis/Hersh

16. An (infinite) series is   a sequence of numbers with plus signs between these numbers.
Bers

17. We have an (infinite) series if, between each two terms of an infinite sequence, we insert a plus sign.
Maak

18. An (infinite) series is   an ordered pair   {a_n}; {s_n}   with   s_n short for a₁ + a₂ + … + a_n
Buck, Gaughan, Maurin, Protter/Morrey, Zamansky, Encyclopaedia of Mathematics1992, Wikipedia-Dutch, Wikipedia-English, Wikipedia-French
Buck writes(1956,1965, 1978): "An infinite series is often defined to be 'an expression of the form Σ₁^∞ a_n '.   It is recognised that this has many defects."

19. If we try to add the terms of an infinite sequence   a   we get an expression of the form   a₁ + a₂ + a₃ + ···   which is called an (infinite) series.
Stewart

20. If we add all the terms of an infinite sequence, we get an (infinite) series.
De Gee

21. When the terms of a sequence are added, we obtain an (infinite) series.
Croft/Davison

22. When we wish to find the sum of an infinite sequence <a_n> we call it an (infinite) series and write it in the form
a₁ + a₂ + a₃ + ···
Keisler

23. Given a sequence a , then the sequence   a₁,   a₁ + a₂,   a₁ + a₂ + a₃, ···   is called an (infinite) series.
Apostol, Burrill/Knudsen, Endl/Luh, Fischer, Forster, S R Lay, Rosenlicht, Wikipedia-Italian

24. Given a sequence  a, then the sequence   a₁,   a₁ + a₂,   a₁ + a₂ + a₃, ···   is called the (infinite) series connected with the sequence a.
Barner/Flohr, Friedemann,
Dijkstra cs (Twente University), Almering (Delft University)

25. Given a sequence  a, then the infinite sum   a₁ + a₂ + a₃ + ···   is called an (infinite) series.
Grossman, Leithold

26. Given a sequence  a, then the expression   a₁ + a₂ + a₃ + ···   is called an (infinite) series.
L J Adams/White, Blatter, Van der Blij/Van Thiel, Gottwald/Kästner/Rudolph, Sze-Tsen Hu

27. Given a sequence  a, the symbolic expression   a₁ + a₂ + a₃ + ···   we call an (infinite) series.
Rudin, Walter

28. Given a sequence  a, an expression of the form   a₁ + a₂ + a₃ + ···   is an (infinite) series.
Thomas/Finney

29. No explicite attempt is made to describe the meaning of (infinite) series, although this term is used frequently.
Ackermans/Van Lint, Binmore, Cheney, Godement, Hille, Hirschman, Johnson/Kiokemeister, Knapp, Kreyszig, Larson/Hostetler, Lax, Morrill, Neill/Shuard, Riley/Hobson/Bence, Van Rootselaar, Ross, Varberg/Purcell/Rigden, Widder, Wikipedia-German, Duistermaat (Utrecht University), D&I (Groningen University)

30. For any sequence $a$ ,   the associated (infinite) series is defined as the formal sum (expression describing a sum) a_M + a_M+1 + a_M+2 + ···  .
Wikipedia-Dutch (fall 2015)

31. An infinite sequence of real numbers is called (infinite) series.   Original wording: On appelle 'série' une suite indéfinie de quantités (quantité: nombre reel).
C.-A. Cauchy.

32. (Infinite) series is the historical name for the mathematical object now mostly called sequence.
C.F. Gauss (1777-1855, Werke Abt.I, Band X, S.400) emphasizes with  "Die Convergenz einer Reihe an sich ist also wohl zu unterscheiden von der Convergenz ihrer Summirung ....", that you should be aware of the fact that in connection with sequences, the word 'convergence' is used in two different meanings. (The convergence of the sequence itself has to be distinguished from the convergence of its summation.)

This not very satisfactory situation, caused by the double meaning of 'convergence' in the 19th century, can be structured by accepting that:
- when 'series' is used denoting a mathematical object, it is synonym with 'sequence' (as in the 19th century and later), and
- in other cases 'series' is designating a certain kind/type of expression (or representation, or evaluation, or maybe even more).
Instead of 'series expression' mostly the shorter 'series' is used.  But one has to realize that with 'convergent series' is not meant: 'the convergent mathematical object named series ', but: the convergent mathematical object denoted by the (type series) expression.
--

@Sławomir Biały.   Please, present one or more explicit examples of occurrences of "antagonistic text" in my posts on this Talk page.  And one or more examples of occurrences of "rambling text" in my posts on Talk page.
I hope I can learn from your examples, how to improve the presentation of my arguments. And how to avoid unnecessary blocking. --

Additional secondary source

To the list of  "Secondary sources supporting Hesselp's edits" (22:52, 27 April 2017, answering Wcherowi's remark 17:16, 25 April 2017  "...your edits are not supported by citations to reliable secondary sources...")  I add:
- R. Creighton Buck (1920-1998, University of Wisconsin),   Advanced Calculus, 1st ed. 1956, 2nd ed. 1965, 3rd ed. 1978 "An infinite series is often defined to be  'an expression of the form Σ₁^∞ a_n '.   It is recognised that this has many defects."
--

Spivak: editions 1967, 1980, 1994, 20??(nog controleren) "........an acceptable definition of the sum of a sequence should contain, as an essential component, terminology which distinguishes sequences for which sums can be defined from less fortunate sequences."

D.Lazard, 15:14, 30 April 2017(UTC)
...a series is a mathematical object. It appears that this concept is not a simple one, as it involves the concept of infinity, which was not well understood nor well accepted before the end of the 19th century (this make your citation of Cauchy irrelevant for discussing the modern terminology; note that he avoided carefully to talk about infinity).

Wikipedia "Mathematical object": A mathematical object is an abstract object arising in mathematics. .... In mathematical practice, an object is anything that has been (or could be) formally defined, ...

Victor J. Katz, A History of Mathematics An Introduction (reprint November 1998), p.705
It was Augustin-Louis Cauchy, the most prolific mathematician of the nineteenth century, who first established the calculus on the basis of the limit concept so familiar today. Although the notion of limits has been discussed much earlier, even by Newton, Cauchy was the first to translate the somewahat vague notion of a function approaching a particular value into arithmetic terms by means of which one could actually prove the existence of limits. Cauchy used his notion of limit in defining continuity (in the modern sense) and convergence of sequences, both for numbers and of functions. .......

CBM

A few comments

I have no desire to enter long discussions about this article, but I wanted to leave a few comments about this revision [14]:

• Articles should be about mathematical objects, not directly about the words for them. So we avoid writing "The word 'group' is used to mean ..." or "The word 'series' is used to mean ..." whenever possible. Instead we write "A group is ..." or "A series is ...". There is another example of this at WP:ISATERMFOR. Similarly, the title (and section) "Situations in which the word 'series' is used" is too focused on the word series instead of the concept.
• Remarks such as "No sources are found, presenting a non-contradictory description of such a mathematical notion, ..." come across as the opinion of an author rather than as encyclopedia-worthy knowledge. Our articles should not assert that all existing sources are contradictory. More likely, when someone claims that all existing sources are wrong, that person has misunderstood something or is promoting an unusual viewpoint.
• The section "Definitions, common wordings" is not, in my opinion, written in ordinary mathematical prose. The spacing in "R e d u c t i o n o f s u m s a n d p r o d u c t s" is out of place and doesn't match any common style on Wikipedia. More generally, the style of the top few sections has too many odd spacings, too many lists and bullets, and does not read as ordinary prose. To the largest extent possible, Wikipedia articles should follow the conventions of all of other mathematical prose.

— Carl (CBM · talk) 15:44, 8 May 2017 (UTC)

@Carl. Thank you very much for your concrete comments.
On point 1:   I understand your remark. But......in this case? You add: "whenever possible". Here we have a mathematical object: (in modern words) a mapping on N. The traditional word for what later on is normally named "sequence". And we have a mathematical concept(?), a certain type of expression (a sign for the 'infinite summation function' plus a sign for a sequence as its argument). You may change the order of the two.   The same 'series-type' we meet when classifying representations (for numbers or functions), and when classifying expansions (for functions).
I'm afraid this cannot be combined in one phrase. I explained this in my article text.

On point 2:   I plan to smooth the content of this footnote. Maybe omit it completely. You are right that this sharp, maybe exaggerated wording is better suited for a discussion on Talk page.

On point 3:   On the unusual spacing in R e d u c t i o n o f . . . you're 100% right, I was lazy when I copied it from elsewhere. On the use of other extra spacings: you cannot see them as making the text, and the formulas, better readable? Enough to accept some deviation from standard style?
And on the use of more 'ordinary prose': maybe a question of taste as well. I shall reconsider this. I wouldn't take as an example the present text of the article. For me that's very far from any encyclopedic style. --

Eppstein

You can be right that I've said/written "an expression (even an infinite expression) cannot be a mathematical object". Please, specify in which post (so, in which context) I wrote this. (To be precise: you didn't say that I wrote this, but that you suppose that I think this.)
Let me say this on it. I'm quite convinced that, in order to have a good idea about what mathematics is and how it works, you should distinguish between the mathematical object 'itself', and the way it is (or: can be) expressed. (Expressed by written mathematical symbols, by written text, or verbally.) In this sense I see mathematical objects as different from expressions.   But, when 'expression' is seen as 'a string of discernible signs, you can study such strings extensively; so in this context such string-expressions can be called 'mathematical objects' with good rights. Is this what you meant with your remark?

You refered to "even an infinite expression".  In several attempts to define a concept 'series' I met this label 'infinite expression'.  But it remains unclear for me which condition should be fulfilled for an expression to be an infinite expression. Can you discern, infinite expression or not? :
a)   $\sum _{n\geq 1}n^{-2}$       b)   $\int _{1}^{2}x^{-2}{\rm {d}}x$       c)   $\sum _{n\geq 1}3\cdot 10^{-n}$       d)   $1\div 3$ .   --

More precise terminology

You refered to "even an infinite expression".  In several attempts to define a concept 'series' I met this label 'infinite expression'.  But it remains unclear for me which condition should be fulfilled for an expression to be an infinite expression. Can you discern, infinite expression or not? :
a)   $\sum _{n\geq 1}n^{-2}$       b)   $\int _{1}^{2}x^{-2}{\rm {d}}x$       c)   $\sum _{n\geq 1}3\cdot 10^{-n}$       d)   $1\div 3$ .   -- Hesselp (talk) 09:38, 9 May 2017 (UTC)

None of those use infinitely many symbols. They are all finite expressions. However, some of them describe series (which if you like you can think of as infinite expressions); for instance (a) would usually be understood as referring to the series $1+{\frac {1}{4}}+{\frac {1}{9}}+{\frac {1}{16}}+{\frac {1}{25}}+\cdots$. This is not in principle different from the fact that as expressions $1+1$ and $2$ are different but that as numbers they are equal. —David Eppstein (talk) 15:43, 10 May 2017 (UTC)

I add two more expressions:   e)  $\sum _{n=1,2,3,\cdot \cdot \cdot }3\cdot 10^{-n}$    f) $1^{-2}+2^{-2}+3^{-2}+\cdot \cdot \cdot$  and ask you to answers on:
A.   Are the expressions labeled e and f  finite expressions as well?
B.   Which out of a - f are referring to a series?
C.   Do you see "referring to a series" as meaning the same as "denoting a series" ?
D.   Please, show an example of an infinite expression . --

David Eppstein

The facts, in short:   David Eppstein was 'baffled' (Talk page 22:01, 8 May 2017) by my incomprehension regarding the true nature of "expressions" and "infinite expressions" (being the central key-term in the definition of 'series').  After asking for the difference between finite and infinite expressions (09:38, 9 May, again 08:44, 10 May), the answer (14:36 and 15:43) was unclear to me, so I made my question more concrete (points A-E, 18:49, 10 May).  Reaction by David Eppstein: "...no more interaction with you", "I see your edits as tendentious and disruptive" and some more not very positive remarks. --

On D.Lazard's post in WP:ANI - 23:35, 10 May 2017

D.Lazard's post in WP:ANI is copied here, in three parts with comments by Hesselp indented.

Hesselp's version of series (mathematics) begins by "In mathematics (calculus), the word series is primarily used as adjective ...". This is not only WP:OR but also blatantly wrong: It suffices to look at any modern textbook of calculus to know that "series" is primarily used in mathematics as a noun.

On "..any modern textbook.." :   For a survey of attempts to define 'series', see the list '32 attempts' in this post. The 32 different wordings can be combined to a handful of really different content. Most of the about 80 authors say that a series IS an expression, but leave it to the reader to find out what's the character of the mathematical object, denoted (described, referred to) by this expression.  The same is the case with the 'definition' in the present version of the article; implying the self-referring sentence:  A series is an expression of the form  ..+..+..+  ···  denoting a series.
And left to the reader as well is the question of how to interprete the label "convergent series". A convergent expression seems to be nonsense, but without any idea about the content of the expression, it's not easy to understand what's really denoted by this label.

In some sources (Spivak, Buck, Dijksterhuis, Van Rooij, Cauchy, Gauss) can be found more explicitely how to interprete the usual wordings. Making it possible to see the connection between the traditional - self-referring - wordings in most books on calculus, and the way how the label 'series' is used by mathematicians in practice.
Only a minimal change in interpretation is needed.  That is: don't say: 'series'  IS the expression  ..+..+..+  ···   itself,  but say: 'series' is a certain TYPE OF expression.  The type, constituted by a summation symbol (the sigma-sign, or the repeated pluses and end-dots) combined with the name of a sequence.
This is what should be an improvement of the article, with its consequences in the wording of the remaining standard content. Helping the reader to grab the meaning of the on-first-site strange combination 'convergent series' (= convergent expression).

Why not OR:   The explanation of the meaning of 'convergent series' - as being nothing else as summable sequence - is the very first statement in chapter 'Series' in Michael Spivak's well known "Calculus".  Already for half a century: 1st ed. 1967, 4th ed. 2008.

Note also that, although series are studied in most textbooks of calculus the only source for Hesselp's lead is about 150 years old (and also misunderstood).

"The only source...."?   No, all 80 rather modern calculus books in this list served as sources. And of the 19th century sources are mentioned earlier: Cauchy, Susler, Itzigsohn, Gauss, Von Mangoldt.   Why doesn’t D.Lazard mentions which one of this five he has studied, and which point in it I should have misunderstood?

The remainder of Hesselp's version of the article continues in the same style and consists only of Hesselp's own inventions, beliefs and/or misinterpretation of the rare source that he produces. D.Lazard (talk) 23:35, 10 May 2017 (UTC

Without concrete examples, I can't comment on this.  Is it the conclusion of everyone who have read this edit? --

Tsirel

Now we observe another attack toward "Series (mathematics)" (see "Relevant discussion at WP:ANI" above); User:Hesselp insists on a single definition of a series as a sequence (of terms). For now the article defines a series as (a special case of) an infinite expression. Another equivalent definition in use is, a pair of sequences (terms, and partial sums). Regretfully, this case is not covered by my "bastion", since the set of series is itself not quite an instance of a well-known mathematical structure (though some useful structures on this set are mentioned in our article). And still, it would be useful to write something like A person acquainted with series knows basic relations between terms and partial sums, and does not need to know that some of these notions are "primary", stipulated in the definition of a series, while others are "secondary", characterized in terms of "primary" notions. Implementation need not be unique. When several implementations are in use, should we choose one? or mention them all "with due weight"? or what? Any opinion? Boris Tsirelson (talk) 16:40, 12 May 2017 (UTC)

@Tsirel.   Five remarks.
a.   Mentioning different worded - equivalent - definitions in "Series (mathematics)" :  no objections from my side.   Provided the wording is logically consistent and complete.
b.   To be able to judge to which extend the 'infinite-expression' version satisfies this condition, the notion infinite expression should be clear first: the link to Infinite expression is not sufficient. See Talk:Infinite expression, and the unanswered questions A-E in Talk page 18:49, 10 May 2017.
c.   Moreover, as every expression,  also an infinite expression should refer to some (mathematical or non-mathematical) object.   The 'infinite-expression' version leds to the self-referreing: "A series is an infinite expression.... denoting a series."
d.   On: "User:Hesselp insists on a single definition of a series as a sequence (of terms)."
Not at all. See section Definition in this edit.
e.   The Bourbaki-definition (series = the couple: sequence; its sum sequence) refers to the former (also Cauchy's) meaning of 'series':  a sequence of terms allowing partial sums. --

Lazard 13 May

• I agree with CBM that an action by an uninvolved administrator is needed, I suggest a permanent ban. In fact, Hesselp has shown many times that he is unable or unwilling to have a constructive discussion. The new edit war quoted by CBM is a new example. It should be noted that the object of this edit war (in which I am not involved) is presented as an answer to my above post of 10 May 2017. In this alleged answer, the main point of my post (the fact that "series" is not an adjective) is not discussed. Instead, he pretends discussing the present content of the article, but, in fact he discusses formulations that never appeared in the article and are invented by him. For example "The same is the case with the 'definition' in the present version of the article; implying the self-referring sentence:  A series is an expression of the form  ..+..+..+  ···  denoting a series. This seems a quote, but the word "denoting" does not appear in the article. This method of changing the wording of the content that he pretends discussing is systematic. This strongly suggests a bad faith; in any case it is definitively impossible to have a constructive discussion with this editor. Therefore, a permanent ban seems the only acceptable solution. D.Lazard (talk) 21:18, 12 May 2017 (UTC)

No criterion for finite / infinite expressions

The question whether or not a sound criterion exists to decide between finite expression and infinite expression is mentioned in the following posts: 21:50 2 May 2017, 09:38, 9 May 2017, 15:43, 10 May 2017, 18:49, 10 May 2017, 20:45, 10 May 2017, 22:19, 10 May2017.
No clear answer on this question is formulated yet.

(moet nog afgemaakt)

Indented my comment on two copied sentences (D.Lazard, 21:30, 12 May 2017):

D.Lazard: In this alleged answer, the main point of my post (the fact that "series" is not an adjective) is not discussed.
      By "main point"  is referred to:
      "Hesselp's version of series (mathematics) begins by "In mathematics (calculus), the word series is primarily
        used as adjective ...". This is not only WP:OR but also blatantly wrong:"
      There is some distance between   "series" is (not) an adjective   and   the word series is primarily used as adjective.
      This "primarily used as" is what I try to illustrate in all my posts.

D.Lazard: For example "The same is the case with the 'definition' in the present version of the article; implying the self-referring sentence:  A series is an expression of the form  ..+..+..+  ···  denoting a series. This seems a quote, but the word "denoting" does not appear in the article.
      Current version of the article, sentence 8:   "Such a series is represented (or denoted) by an expression... ".
      Reading backwards: "The expression ... denotes (or: is denoting) a series." --

Comments on D.Lazard's post 23:35, 10 May 2017 :
On "..any modern textbook.." :   For a survey of attempts to define 'series', see the list '32 attempts' in this post. The 32 different wordings can be combined to a handful of really different content. Most of the about 80 authors say that a series IS an expression, but leave it to the reader to find out what's the character of the mathematical object, denoted (described, referred to) by this expression.  The same is the case with the 'definition' in the present version of the article; implying the self-referring sentence:  A series is an expression of the form  ..+..+..+  ···  denoting a series.
And left to the reader as well is the question of how to interprete the label "convergent series". A convergent expression seems to be nonsense, but without any idea about the content of the expression, it's not easy to understand what's really denoted by this label.

In some sources (Spivak, Buck, Dijksterhuis, Van Rooij, Cauchy, Gauss) can be found more explicitely how to interprete the usual wordings. Making it possible to see the connection between the traditional - self-referring - wordings in most books on calculus, and the way how the label 'series' is used by mathematicians in practice.
Only a minimal change in interpretation is needed.  That is: don't say: 'series'  IS the expression  ..+..+..+  ···   itself,  but say: 'series' is used to label a certain TYPE OF expression.  The type, constituted by a summation symbol (the sigma-sign, or the repeated pluses and end-dots) combined with the name of a sequence.
This is what should be an improvement of the article, with its consequences in the wording of the remaining standard content. Helping the reader to grab the meaning of the on-first-site strange combination 'convergent series' (= convergent expression).

Original Research ?:   The explanation of the meaning of 'convergent series' - as being nothing else as summable sequence - is the very first statement in chapter 'Series' in Michael Spivak's well known "Calculus".  Already for half a century: 1st ed. 1967, 4th ed. 2008.  See More precise terminology 21:37, 9 May 2017

"The only source...."?   No, all 80 rather modern calculus books in the list in this post, 20:28, 8 May 2017 served as sources. And of the 19th century sources are mentioned earlier: Cauchy, Susler, Itzigsohn, Gauss, Von Mangoldt.   Why doesn’t D.Lazard mentions which one of this five he has studied, and which point in it I should have misunderstood?

The remainder of Hesselp's version....  Without concrete examples, I can't comment on D.Lazard's last sentence.  Is it the conclusion of everyone who have read this edit? --

Citations, observations, supposition

Attempting to find a way to some kind of consensus, I add the following lines to this Talk page.

Citations, taken out of longer posts on Wikipedia talk:WikiProject Mathematics
- Tsirel - 19:15, 12 May 2017: ".. in general an expression has no value (but in "good" cases it has);"   (Comment Hesselp: the dispute is about the question whether a series-type expression has (in "good" cases) a number as its value, or a series (For: "a series is denoted by an expression like ..+..+..+···"))
- CBM - 20:00, 12 May 2017: "... the definitions that are often given in the books lack something that would be present in a graduate level text."   (Comment Hesselp: No one has presented such a graduate level text in this Talk page.)
- CBM - 20:00, 12 May 2017: "...we should follow the sources and present the same general understanding that they convey.]   (Comment Hesselp: That's easier said than done, see survey in 09:38, 9 May 2017)
- CBM - 20:09, 12 May 2017: "If numerous sources all find it possible to discuss a concept without a formal definition, we can certainly do so as well."
- D.Lazard - 20:43, 12 May 2017: " In any case, a series is not a sequence nor a pair of sequences nor an expression. It is an object which is built from a sequence."   (Comment Hesselp: D.Lazard's edited since 09:50, 14 Februari 2017 seven times a version with:  "a series is an expression").
- Tsirel - 05:02, 13 May 2017: "What does it mean? A vague term whose meaning is determined implicitly by the context, case-by-case?"
- Taku - 23:10, 13 May 2017: "... a series is a more of a heuristic concept than an explicitly defined concept."

Observations   Studying the terminology used in the 19th (and a good part of the 20th) century, concerning the 'series-representation' of numbers (and of functions), we can see two noteworthy points.
(1) The word 'series' was used frequently in situations where we should use 'sequence' now. (Also German 'Reihe' in 'Folge'-situations, and French 'série' in 'suite'-situations.)   Cauchy introduces 'série' explicitely for a sequence with numbers as terms; much later Bourbaki seems to copy this by using 'series' for a sequence with terms allowing the existence of a 'sum series'. The names 'arithmetical series', 'harmonical series', 'Fibonacci series', etc. were in common use. (2) The words converge/convergent/convergence were used in case the terms have a limit, as well as in case the partial sums have a limit. Cauchy seems to use the verb 'converger' for terms with a limit, and the adverb 'convergent' for partial sums with a limit; quite confusing.   And Gauss once remarks: (Werke Abt.I, Band X, S.400) "Die Convergenz einer Reihe an sich ist also wohl zu unterscheiden von der Convergenz ihrer Summirung ...." (The convergence of the sequence itself has to be distinguished from the convergence of its summation.)

Suppostion   This situation: two words (series and sequence) for one notion, and one word for two properties (limiting terms and limiting partial sums), caused ongoing confusion. More and more culminating in a belief in the existence of a third 'mathematical object', apart from 'sequence' and 'the sum sequence of a given sequence'. A mysterious object or notion, whose definition/description causes the difficulties mentioned in the citations above.
How about the idea of describing this historical roots of the present confusion, in the Wikipedia article?  Can this be seen as a description of the existing situation, or is this seen as OR? --

CBM

I don't think the article should focus on the historical roots to any great extent, except perhaps in a section on history. Sources from the 19th century are not likely to be of much use in this kind of elementary article, and indeed there were many more terminological problems at that time (compare the common use of "infinitesimal" at that time). Every contemporary calculus book I have seen has the same concept of a series, although of course the wording may vary from one author to another. — Carl (CBM · talk) 01:21, 15 May 2017 (UTC)

I.   On the contemporary calculus books you have seen:

- You have seen Stewart ?   "If we try to add the terms of an infinite sequence, we get an expression of the form  ..+..+..+ ···  which is called an (infinite) series."   (Every time I try to add the terms of an infinite sequence I get - after some hours - a heavy headache, not a 'series')
- You have seen Spivak ? A sequence is called summable if its sum sequence converges. This terminology is usually replaced by less precise expressions.
- You have seen the 'Bourbaki'-followers: Buck, Gaughan, Maurin, Protter/Morrey, Zamansky, Encyclopaedia of Mathematics1992, Cauchy ?  A sequence with an existing sum sequence, is called a series.

The following wordings (taken from calculus books 1956 - 2008) are describing the SAME CONCEPT ?   How many readers of Wikipedia can 'see this cat' ?
- An (infinite) series IS   an expression of the form  ..+..+..+ ···
- An (infinite) series IS   a formal infinite sum.
- The formal expression  ..+..+..+ ··· IS CALLED an (infinite) series.
- An (infinite) series is   a sequence
- An (infinite) series is   a sequence  whose terms are to be added up.
- An (infinite) series is   the sum of the terms of a sequence.
- An (infinite) series is   an infinite addition of numbers.
- An (infinite) series is   a mathematical proces which calls for an infinite number of additions.
- An (infinite) series is   a sequence of numbers with plus signs between these numbers.
- We have an (infinite) series if, between each two terms of an infinite sequence, we insert a plus sign.
- An (infinite) series is   a sequence, followed by its sum sequence.
- An (infinite) series is what we get if we add all the terms of an infinite sequence.
- When we wish to find the sum of an infinite sequence we call it an (infinite) series
- The sum sequence of a given sequence is called an (infinite) series.
- The sum sequence of a given sequence is called the (infinite) series connected with the given sequence.

To CBM and others:   Present the mean value of LCM and GDC of this 15 wordings.

II.   Can you mention one or more titles (of calculus books you have seen) with a definition / description of "series",  NOT self-referring - explicitely or implicitely - with phrases like:
• a series is an expression of the form  ..+..+..+ ···,   combined with
• the expression  ..+..+..+ ···  refers to (denotes) a series. ? --

III.  @CBM:  In your edit summary Article 01:29, 15 May 2017 you emphasize:  ..the key definition up front, which needs to move directly to the SUM of a series.." .
Isn't that exactly the content of the fist few sentences of this edit ?   As that lines try to say:

The (series-type) expression       (with symbols for the summation-function, and for a sequence as its argument)
denotes / refers to       (in case of a valid - not a void - expression;  the "good" ones, Tsirel says)
the SUM number of the named sequence.       (or the SUM function in case of function terms)
(So now the expression ..+..+..+··· is not cycling back to "series" again.) --

CBM - 16 May

- Instead of the heading "Definition", I have in mind: "Names and notations".

- About recent changes in the text of the article:
• The self-referring "A series is an expression denoting a series" can't be found in the text any longer. Improvement.
• In the definition of 'series', the two-track construction "a series is an infinite sum, is an infinite expression of the form .." disappeared. Improvement.
• The "such as" regarding the capital-sigma notation. Improvement. (Maybe some more variants can be shown? As well as
  a1 + a2 + ... + an + ...  as variant of the pluses-bullets form.)
• The label "infinite expression" (instead of "expression") is still there. Although no criterion is found for decerning. See
  $\sum _{n\geq 1}n^{-2}$  ,  $\sum _{n=1,2,3,\cdot \cdot \cdot }3\cdot 10^{-n}$  ,     $1\div 3$  .
• The intro (almost at the end) says:  "When this limit exists, one says that the series is convergent or summable, and the limit is called the sum of the series.   And the present definition says:  "a series is an infinite sum,..". Combined we get wordings as:  "a summable infinite sum"  and  "the sum of an infinite sum".
I know there are books where you can find this; but it's not very nice and comprehensible.   Is it definitely OR to add that it's not unusual to say "summable sequence" and "sum of a sequence" as well?  I referred to Spivak (1956...2008) and many hits in Google.

The third sentence in the present text says: "Series are used in most areas of mathematics,..".   Isn't it true that the content of this sentence can be worded as well by:  "Capital-sigma expressions and pluses-bullets expressions are used in most areas of mathematics".
Why are this notations so important? Because they express a method to denote/describe irrational numbers (and as an generalization also functions) by means of a regular-patterned sequence with more familiar rationals as terms (or 'easier' functions).
The usual word for such a method to describe mathematical objects by means of simpler objects, is "representation".   We have: the decimal representation, the continued fraction representation, the infinite product representation, and some more. Not the least important is, what could be called "the infinite sum representation" or - in honour of the famous term - "the series representation". The representation based on the summation function for infinite sequences.
So, instead of saying "series are important" (with the hard to define term 'series'), you could say "the series representation is important" (describable without mysterious words). Is this a so big change that you are going to react with: "impossible, clear OR" ?

Last remark. Caused by personal circumstances I've to tell that I leave by now Wikipedia for at least a couple of weeks. I wish you fruitful discussions. Hessel Pot --