Talk:Series (mathematics)

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 .

This reacting on attempts made to start discussion on the content, 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)

-- Hesselp (talk) 10:20, 6 May 2017 (UTC)

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, editions 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."
-- Hesselp (talk) 13:42, 8 May 2017 (UTC)

"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 attempts

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 expansion, 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.
-- Hesselp (talk) 20:28, 8 May 2017 (UTC)

Like Carl below I don't want to get into a long discussion. But let me just take this opportunity to register my bafflement that you think an expression (even an infinite expression) cannot be a mathematical object. —David Eppstein (talk) 22:01, 8 May 2017 (UTC)

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)   $1\div 3$ .   -- 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 three more expressions:   e)  $\sum _{n=1,2,3,\cdot \cdot \cdot }3\cdot 10^{-n}$    f)  $\sum _{n=1}^{\infty }n^{-2}$     g) $1^{-2}+2^{-2}+3^{-2}+\cdot \cdot \cdot$  and ask:
A.   Are the expressions labeled e, f, g  finite expressions as well?
B.   Which out of a - f are usually understood as referring to a series (describing 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.
E.   In the present text "series" is defined as being: an infinite expression (of a certain type). You write "If you like you can think of a series as being an infinite expression (an infinite tree labeled with symbols of various types)." That's not a definition as we are used to in mathematics, isn't it? -- Hesselp (talk) 18:49, 10 May 2017 (UTC)

Like other participants here, I don't see the point in continuing to participate in interactions with you. Your discussion here does not seem to be based on improving the article based on mainstream mathematical work, but about some sort of nitpicky definitional dispute which is mostly off-topic for this article. —David Eppstein (talk) 19:17, 10 May 2017 (UTC)

It was you, David Eppstein, who mentioned 'infinite expressions' (the central key-word in the Definition-section in the present text) in your first post (22:01, 8 May 2017), registering your bafflement about my ignorance.  Now, asked for an example......you are away!
I repeat my questions A-E.   Somebody else?   Again, this is about the heart of the article - the definition. -- Hesselp (talk) 20:45, 10 May 2017 (UTC)

Your repeated faux-naive questions don't address my earlier comments. Instead, they make clear that you are not gaining any understanding out of this interaction, are not working towards clarification of the article, and are merely continuing to try to win points, trip up other editors, and push your idiosyncratic viewpoint. What is the point of playing that game? —David Eppstein (talk) 20:55, 10 May 2017 (UTC)

'don't address my earlier comments' ? See your posts 14:36 and 15:43, 10 May 2017.
And yes, I try to win the point that the present 'definition' is at best a self-referencing sentence. I proposed an alternative wording.  I don't see why that couldn't be seen as an attempt to improve/clarify the article. Questions A-E are still unanswered..... -- Hesselp (talk) 22:19, 10 May 2017 (UTC)

More precise terminology

In this edit Michael Spivak's Calculus (editions 1967, 1980, 1994, 2008) was added as reference, with as edit summary: "Citing a standard calculus text is sufficient to verify all content of these sections.".
Spivak advocates as  more precise  terminology over the  somewhat peculiar  standard language (using 'series'):
- summable sequence
- sum of a sequence
- absolutely summable sequence
- uniformly summable sequence
- Cesaro summable sequence
- Abel summable sequence
This terminology could help to clarify expressions with 'series' in the article as well. --Hesselp (talk) 21:37, 9 May 2017 (UTC)

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 [1]:

• 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? Fortunately 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. -- Hesselp (talk) 22:17, 8 May 2017 (UTC)

@Carl. Once more on your point 2.
Footnote 2, extending text-sentence 11 in [2], says:  "No sources ARE FOUND....OF SUCH a notion."   You refer to this words with ALL EXISTING SOURCES are contradictory/wrong. That's not the same. No reader can expect that an article is written by people who have studied ALL existing sources. I'll consider how to prevent misreading at this point. --Hesselp (talk) 16:17, 9 May 2017 (UTC)

Your excuse for the unusual spacing is troubling to me. You should not be copying text here from elsewhere. —David Eppstein (talk) 21:50, 9 May 2017 (UTC)

@David Eppstein. Don't be worried or troubled. I copied these lines from this post. The 'unusual' spacing I used the day before in this post as well, to make headings in a proposal for a longer edit (not meant as sector-headings in Talk page).
I understand that it is not easy to find the condition for an expression to be an infinite expression ? --Hesselp (talk) 08:44, 10 May 2017 (UTC)

How, not easy? It's just an infinite tree labeled with symbols of various types. —David Eppstein (talk) 14:36, 10 May 2017 (UTC)

David Eppstein, if you haven't seen this discussion you may want to read it. I don't believe it is worth responding in any way to Hesselp's numerous posts to this talk page unless they post something that looks like it might actually gain consensus. Mike Christie (talk - contribs - library) 15:22, 10 May 2017 (UTC)

I did see it, but thanks for the reminder. —David Eppstein (talk) 15:36, 10 May 2017 (UTC)

Equivalent definitions, again

See Wikipedia talk:WikiProject Mathematics#Equivalent definitions, again. Boris Tsirelson (talk) 16:47, 12 May 2017 (UTC)

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 ? -- Hesselp (talk) 20:20, 14 May 2017 (UTC)

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. ? -- Hesselp (talk) 16:53, 15 May 2017 (UTC)

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.) -- Hesselp (talk) 17:43, 15 May 2017 (UTC)

Well, yes, I think there is a single concept of "series" that all these books are presenting. On the other hand, a long "wall of text" is not pleasant to read on a talk page, and I am not likely to continue reading them. Please take the time to express yourself succinctly. I don't find the version that describes "series" as an adjective to be particularly compelling. Actually, I don't see anything exceptionally faulty with the current phrasing of the article, which I have just read again. Of course everyone has their own way of saying things, I and I would write things differently if it was my article, but I don't see any deep issues at the moment. I do plan to do some more copyediting over time. — Carl (CBM · talk) 18:39, 15 May 2017 (UTC)

I, for one, am mostly disturbed by a single word in this article: the title "Definition" of Sect. 1.1. I understand that we have good reasons not to give a single (up to equivalence) mathematical definition. But in a mathematical article (even undergraduate) I would not call "definition" something that is not a mathematical definition. I'd better inform the reader shortly but honestly, why no definition. Such words as "definition", "theorem" and "proof" are somewhat sacred for me.Boris Tsirelson (talk) 19:05, 15 May 2017 (UTC)

I have similar misgivings about the word "Definition". Perhaps it is appropriate to point out that a series ("the sum of infinitely many terms") is a mathematical concept that does not have a generally agreed upon definition, just as "the area under a graph" is a concept that does not have a proper mathematical definition, but can be formalized in different ways depending on the circumstances. Sławomir Biały (talk) 20:25, 15 May 2017 (UTC)

Saying that there is "no" definition may be too strong, and some might even claim it would be "original research". Perhaps we could simply remove the subheading "Definition", or change it to a different word. — Carl (CBM · talk) 23:25, 15 May 2017 (UTC)

Carl, you are the first among us to know exactly the meaning of "definition", "theorem" and "proof" in mathematics. Boris Tsirelson (talk) 05:10, 16 May 2017 (UTC)

I agree that the header "Definition" has to be changed. IMO, this should accompanied by some restructuring of the article. I see a first section "Motivation" for regrouping the details of Achilles-and-the-tortoise paradox and other explanations (this would allows reducing the size of lead by replacing the corresponding paragraph of the lead by a single sentence), and a second section "Basic properties", which should be rewritten for avoiding too much repetitions of the content of the lead. D.Lazard (talk) 08:27, 16 May 2017 (UTC)

- 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 -- Hesselp (talk) 08:53, 16 May 2017 (UTC)

I'm not a mathematician, but I came here confused on this point, so it is not just an advanced point. Is a series the sum of a sequence, or can it also refer to a sequence? Must a series be infinite to be called a "series"? If mathematicians use the terms loosely in contradictory ways, the article should say that, and then say what definition the article is going to use and continue with just that. WHat matters here is not history (though that could be a separate section) but current usage.--editeur24 (talk) 16:49, 18 December 2020 (UTC)

For the record, Hesselp has been indefinitely banned for editing any Wikipedia article on series and related to series, including talk pages. This old discussion is a part of the reasons of this ban. It is thus not surprisingly that it confuses non-mathematicians. The lead has evolved since this discussion, by making clear that a sequence defines a series, but is not the same thing as the series it defines. I think that the best answer to your questions is the following paragraph that I have just added to the article.

The notation $\textstyle \sum _{i=1}^{\infty }a_{i}$ denotes both the series—that is the implicit processus of adding the terms one after the other indefinitely—and, if the series is convergent, the sum of the series—the result of the process. This is a generalization of the similar convention of denoting by $a+b$ both the addition—the process of adding—and its result—the sum of a and b.

D.Lazard (talk) 17:54, 18 December 2020 (UTC)

Thank you. That addition is helpful, and worth the space, I think. The analog to addition is very good. Maybe the whole intro could use tightening, but if it is, I hope that analogy survives. Maybe a sentence would be useful to the effect of "People sometimes loosely speak of the sequence a_n being summed as the 'series a_n', but that is misuse of the term."--editeur24 (talk) 19:08, 18 December 2020 (UTC)

Please do not add editorializing like that without a reference to a reliable source -- dubious, unsourced editorializing does not belong here. --JBL (talk) 19:17, 18 December 2020 (UTC)

We want Wikipedia to be useful to the novice as well as the expert, and novices will want to know the difference between a sequence and a series. The start of an article seems a good place to clear that up.
I think the confusion over sequence vs. series is common knowledge, rather than something that needs citations. Here are a couple of web cites I found very quickly, but I think inserting them would be more distracting than useful:

"In mathematics and statistics, the line that demarcates sequence and series are thin and blurred, due to which many think that these terms are one and the same thing." https://keydifferences.com/difference-between-sequence-and-series.html

Students do not understand the difference between series and sequence and sometimes pay dearly with their marks being deducted when they use these terms incorrectly. https://www.differencebetween.com/difference-between-series-and-vs-sequence/ — Preceding unsigned comment added by Editeur24 (talk • contribs) 21:57, 18 December 2020 (UTC)

Formal sums

A recent edit by CBM brought my attention to the fact that our article says

While the most common uses of series refer to their sume, it is also possible to treat series as formal sums, meaning that no operations are actually performed, and the symbol "+" is an abstract symbol of conjunction which is not necessarily interpreted as corresponding to addition.

However, elsewhere, we define "formal sums" as members of a free abelian group, and the phrase formal sum redirects to our article on free abelian groups. This differs in several ways from the meaning here, notably that the free abelian group members are sums of only finitely many nonzero elements and that their ordering is unimportant, neither of which is true for formal series. Would it maybe make sense to turn the formal sum into a disambiguation page that points to this article for formal series, as well as pointing to the other article for the other meaning? Are there other meanings than these two that should be considered? —David Eppstein (talk) 04:26, 15 May 2017 (UTC)

Series belong to a completion of the free abelian group. I think the article free abelian group could discuss this, and that may eliminate the need for a disambiguation. (Edit: Ordering is obviously a bigger problem, though.) Sławomir Biały (talk) 12:49, 15 May 2017 (UTC)

Do they really? The free abelian group on what generators, then? —David Eppstein (talk) 14:36, 15 May 2017 (UTC)

1. A formal series is, normally, a formal sum whose terms are indexed by the additive monoid $\mathbb {N} .$ If the terms are numbers or belong to an abelian group, the standard notation becomes ambiguous (should the additions be done or not?). This is usually solved by remarking that the monoid $\mathbb {N}$ is isomorphic with the multiplicative monoid $(1,X,X^{2},\ldots ,X^{n},\ldots ).$ This induces an isomorphism between formal series and formal power series. It follows that "formal series" is a phrase that is rarely used, except as an abbreviation for formal power series.
2. A prototype for the free abelian groups of rank n consists of polynomials with integer coefficients, of degree less than n. As, when n increases the union of these abelian groups is the ring of polynomials, the completion suggested by Sławomir is simply the completion of the polynomial ring for the X-adic topology. That is, the ring of formal power series is the completion of the polynomial ring for this topology.
3. My conclusion is that, in this article, we must avoid to talk about formal sums: With the definition of a series as an infinite succession of additions, talking of formal sums in section Definition is unnecessarily confusing, and, because of above remarks, section Formal series must be replaced by a section "Formal power series". D.Lazard (talk) 16:37, 15 May 2017 (UTC)

Formal series are not usually restricted to having integer coefficients, so the connection to free abelian groups seems specious to me. —David Eppstein (talk) 16:59, 15 May 2017 (UTC)

I think there may be some expositional advantages to merging the section on "formal series" (which I moved from the definition) into the section on power series. Of course not all power series are formal series, but I think the most common use of formal series is via formal power series. Moreover, the section on power series currently mentions generating functions, which is a key topic for formal power series. — Carl (CBM · talk) 17:04, 15 May 2017 (UTC)

To editor David Eppstein: You are right, one has to replace everywhere "free abelian group" by "free module" or "vector space". After all, a free abelian group is a free module over the integers. Thus one looses nothing by this replacement. Moreover, as presented in Formal sum, the phrase "formal sum" is confusing. In fact, as defined there, the abelian group of formal sums over a set S contains the formal differences of two elements of S. D.Lazard (talk) 17:36, 15 May 2017 (UTC)

To editor D. Lazard and/or David Eppstein: I think the point is that it's really the "free algebra", completed in a very strong topology. Sławomir Biały (talk) 02:11, 16 May 2017 (UTC)

I think I have figured out what my point was, more clearly, after all this time. It is that the elements of free abelian groups (or more or less the same thing, free $\mathbb {Z}$-modules) are always finite sums of scaled basis elements. But series, as we usually want to define them, are sums of infinitely many terms. So you can still think of them as being some kind of formal sum if you like, but the type of thing they belong to is not a free abelian group. Saying "free module" does not really make any difference here. Infinite formal sums of elements of a field are definitely elements of a vector space over the field (you can add them and scale them), and in the same way infinite formal sums of elements of a ring form a module but not necessarily a free one. In particular, the infinite formal sums of integers are the Baer–Specker group which was proved by Baer 1937 not to be free. —David Eppstein (talk) 00:28, 11 May 2019 (UTC)

IMO, this article should have a section "Formal power series" which is not a subsection. Beside the definition, this section should contain the fact that there is a bijection between sequences and formal power series; this bijection is used in many parts of mathematics and allows studying deep properties of integer sequences; examples are generating series, in combinatorics, and also Hilbert–Poincaré series in topology, and Hilbert series in algebraic geometry. This could also be useful for clarifying, for the reader, the relationship between sequences and series. D.Lazard (talk) 08:54, 16 May 2017 (UTC)

I have started this process. Please feel free to edit it more, keeping in mind that this should be a more elementary article when possible. I fear that some of the text may already be technical for a general article on series. — Carl (CBM · talk) 15:44, 22 May 2017 (UTC)

Query

The following text in the introduction: 'The resolution of the paradox is that, although the series has an infinite number of terms, it has a finite sum, which gives the time necessary for Achilles to catch up with the tortoise'. Really, is this so? You could reasonably argue that if you were to offer Zeno the sequence, 1/4 + 1/16 + 1/64... he could say that it equals a number that is infinitely smaller than 1/3... -- Knucmo2 (talk) 23:52, 10 May 2019 (UTC)

He might say that -- indeed, I think this is roughly consistent with how the Greeks would have understood infinite sums -- but the point is that according to the rigorous modern understanding of infinite summation, it is actually equal. --JBL (talk) 00:05, 11 May 2019 (UTC)

Proposal: Move content of "Generalizations" section into its own article

I feel that the "Generalizations" section is sufficiently large and well-developed to be its own article. Since most readers of this article are (presumably) students learning about the basics of series, I think that the "Generalizations" sections will not be helpful for them and not used by them. In this new article we'd be able to add as many more technical details as we please without bloating the "Series" article. We'd replace the content of the "Generalizations" section with a paragraph or two that gives a quick summary of the generalizations. I propose the following for the "Generalizations" article: User:Mgkrupa/Generalizations of Series Proposal Mgkrupa (talk) 23:18, 2 June 2020 (UTC)

Please, add new discussions at the bottom of the talk pages. D.Lazard (talk) 08:05, 3 June 2020 (UTC)

The two first subsections of section "generalization" must remain there, as being summaries of existing articles, that are sufficiently close to the subject for being mentioned here. I would not object if someone creates an article Series over an arbitrary index set, and replaces the long section "Summations over arbitrary index sets" by a summary of this future article. I oppose to a move that would not leave a summary here. D.Lazard (talk) 08:20, 3 June 2020 (UTC)

I agreeMgkrupa (talk) 12:18, 5 June 2020 (UTC)

Is the term "finite series" correct usage?

I am not a mathematican. The term "finite series" is often used to mean the sum of a sequence with finite terms. Is this correct? Just "series" seems to be used for an infinite sum (from my Rudin and this article). Whether the term "finite series" should be used or not would be useful item for this article, if there is well-established usage. Ideally, it would have citation in support of whatever is said. --editeur24 (talk) 19:22, 18 December 2020 (UTC)

As far as I know, "finite series" is not used in modern mathematics, but I believe that it was used in the 19th century. The modern terms are "finite summation" and "finite sum", although "finite" is often omitted in both phrases. See Summation. D.Lazard (talk) 21:53, 18 December 2020 (UTC)

Actually, I just checked Google Scholar for "finite series" and there are many articles that use it, including some with hundreds of cites, so they must be respectable. It's very common in an ordinary google search too, in the high school and college math level sites. --editeur24 (talk) 22:43, 18 December 2020 (UTC)

Finite series is just a partial sum. Valery Zapolodov (talk) 12:43, 25 August 2021 (UTC)

Is there a good source to confirm that? If there is, it is worth adding this statement to the article, just a short mention like "partial sum (also sometimes referred to as finite series)". Shcha (talk) 11:18, 26 May 2023 (UTC)

I have found the following definition of series in "Encyclopedia Of Mathematics" by J.Tanton: "=SERIES= A sum of numbers is called a series. The sum could be finite, such as 2 + 4 + 6 + 8 + 10, for example, or it could be an infinite sum, as for 2 + 5 + 8 + 11 + 14 + … for instance. Each number in the sum is called a term of the series."

However many other sources which I checked either define series as an infinite sum, or define infinite series only, or use the term "series" to refer to infinite series only.

Another noteworthy source which refers to finite series is DfE (UK education authority) which in their A Level Mathematics qualification spec mentions "the nth term and the sum of a finite geometric series". That further propagates to some A-level exam preparation resources, including textbooks and revision guides. Shcha (talk) 11:15, 26 May 2023 (UTC)

Just searching for "finite series" or "finite geometric series" in google books demonstrates that finite series is an established concept in modern mathematics, although far less common than "infinite series". Shcha (talk) 11:52, 26 May 2023 (UTC)

"Finite series" is sometimes used, but this does not means that this is a concept. On the contrary, this is an ambiguous phrase. If series is taken in its common-language meaning, then a finite series is nothing else than a finite sequence. On the other hand, a series is, in mathematics, the sum of an infinite sequence, and one may be tempted to call finite series the sum of a finite sequence. But the correct term is sum or summation, and, if finite series is used, it is unclear if it refers to the sequence or to its sum.

So, to answer to the original question, the term finite series is not correct usage, as it is ambiguous, and there are unambiguous mathematical terms for each of its possible uses. D.Lazard (talk) 14:16, 26 May 2023 (UTC)

Contrary to the rather archaic prescriptive view of language which D.Lazard is following, a word means what it is used to mean and understood to mean, neither more nor less. The notion that a word used in a way which in which it widely used and understood is somehow "incorrect" has no objective basis. I personally think it is on the whole better to avoid the use of "finite series", but the subjective view that something is better avoided falls well short of being the same as that thing being in some sense objectively "incorrect". The expression exists, and is used and recognised, in my experience most commonly to refer to the sum of a finite sequence, particularly a finite segment at the start of an infinite sequence. JBW (talk) 14:47, 26 May 2023 (UTC)

If you checked a few books which I gave the links to, you would see that "finite series" is not just "used sometimes", but used quite often, especially in college level textbooks, when students are first introduced to sequences and series. There, series are often defined as a sum of elements of a sequence, and finite series in particular are defined as a sum of elements of a finite sequence. Ambiguous or not, but this is how this term is used sometimes in mathematics, and it certainly deserves to be reflected in the article. I personally don't like that. It's not ambiguous, it's superfluous, as finite series are just regular sums and don't have any special properties unlike infinite series, so they are not really "studied" in mathematics. But this term is used e.g. by DfE, and students will go to wikipedia to figure out what "finite series" is, and it's not very helpful that the article "series (mathematics)" does not cover that in any way. Shcha (talk) 22:00, 4 June 2023 (UTC)

Nominating the article for GA

Hi all, I am planning to nominate the article as a GA. I feel it is well-written, clear and understandable. Any suggestions? Justlookingforthemoment (talk) 19:52, 21 February 2022 (UTC)

Some opinions:

• This is definitely an important enough article to be worthy of getting to GA status.
• I see that you have already been editing the article, but those edits look very minor. Drive-by GA nominations are not taken seriously, so consider putting significantly more editing into the article before nominating. It needs it, anyway; see below. It is not currently in a nominable state.
• The only mention of paradoxes is in the lead. Lead material should consist solely of summaries of material covered in-depth later, so consider moving that material out of the lead and only summarizing it in the lead. Also, it is unsourced. Everything needs a source.
• The entire "Convergent series" section is unsourced, as is the entire "Convergence tests" section, and many other entire paragraphs are unsourced. If I were reviewing this for GA, this lack of sourcing would be enough for me to quick-fail it. So sourcing is the most urgent revision that should be made before the article is nominated. Every claim in the article needs a footnote to a reliable published source. That means, at least, every paragraph that is not merely an introductory summary to material covered in more detail later, and (if the whole paragraph cannot be matched to a single source) the sentences within it.
• The "Examples of numerical series" section is long, formula-heavy, and poorly sourced. Is all of it necessary? Consider Good Article criterion 3b: "it stays focused on the topic without going into unnecessary detail". Same goes for some other material, which might be better summarized briefly with pointers to other articles.
• There are two styles of referencing in use: short reference footnotes with pointers to the bibliography, and long references in the footnotes themselves. Can it be made more consistent?
• Many of the sources are high-quality mathematics textbooks, but some (e.g. MathWorld or the arxiv preprints) are not; consider replacing those ones.
• Good Article criterion 1a does not prevent technical articles from becoming Good Articles, but the part about "appropriately broad audience" does mean that it would be worthwhile to look through the article making sure that none of its technicality is unnecessary. After the sourcing issues and the overall logical structure of the article, that would be my next priority in revision of the article.

—David Eppstein (talk) 20:24, 21 February 2022 (UTC)

"Many of the sources are high-quality mathematics textbooks, but some (... or the arxiv preprints) are not; consider replacing those ones" What? At least look it up on semantic scolar before deleting, as in last change. 2A00:1370:8184:1CE9:49C2:D9DE:CBA1:7279 (talk) 01:17, 6 April 2023 (UTC)

overly dogmatic removal of unpublished preprints?

There’s an ongoing dispute about this recently removed chunk of the article, which should be discussed here (see WP:BRD) instead of in edit summaries of a revert war:

Alekseyev (2011) proved that if the series converges, then the irrationality measure of $\pi$ is smaller (or equal) than 2.5, which is much smaller than the current known bound of 7.10320533.... In 2022 the opposite was also proved, that if the irrationality measure of $\pi$ is smaller than 2.5 it would imply convergence, the case of equal to 2.5 remains unsolved.

The first of these papers has been cited 15–20 times, mostly in other preprints but also including a few times in peer-reviewed papers. The author is an established researcher with many widely cited papers. It doesn’t seem like this is on its way to being published elsewhere, but the result is apparently accepted by others in the field. I don’t know or care too much about this topic, but if this article is going to discuss the Flint Hills series at all, it seems (just from these superficial signals) like a fine citation. Insisting that this cannot possibly be cited because it is a preprint seems like excessive dogmatism at readers'/Wikipedia's expense.

The second paper is by a PhD student at the beginning of their career, and is very recent. That one seems more open to some discussion here, which I will leave to experts in this subject. –jacobolus (t) 03:31, 6 April 2023 (UTC)

The argument here is that both of those sources (and last one I added) are used in https://en.wikipedia.org/wiki/Liouville_number ("It has been proven that if the series..."). I also fixed a typo about less or equal. Oh, and also 2 reverts over 24 hours is not yet an edit war. Valery Zapolodov (talk) 12:11, 6 April 2023 (UTC)

Per WP:USERGENERATED, Wikipedia is not a valid source. So, the content of Liouville number is not an argument here. D.Lazard (talk) 12:33, 6 April 2023 (UTC)

That is only because wikipedia is not WP:RS. So it cannot be used to cite stuff. What I can do though is to ask why it is allowed to remove cites here that are used elsewhere. Where is a genius decision that peer reviewed many times old arxiv papers are somehow not RS? Or that source code is not reliable. Same happened when someone nuked the sources here: https://en.wikipedia.org/w/index.php?title=Sch%C3%B6nhage%E2%80%93Strassen_algorithm&diff=prev&oldid=1147228907&diffmode=source Valery Zapolodov (talk) 13:23, 6 April 2023 (UTC)

off-topic discussion about Schönhage–Strassen sources

The source there is a github bug ticket on what appears at a glance to be an obscure student code project (P.S. the project homepage has an expired SSL cert TLS cert with incomplete chain of trust – not really a good sign) not used in production anywhere. If it was relevant to published work, cite the publication. I don’t think the example there is comparable to the Alekseyev paper under discussion here. –jacobolus (t) 17:47, 6 April 2023 (UTC) "be an obscure student code project" You just called Waterloo Maple an obscure code project?? Are you serious right now? This is literally part of Maple 2021 codebase, different repo same username, and the main BPAS project is done by INRIA: https://github.com/orcca-uwo/MultivariatePowerSeries Valery Zapolodov (talk) 20:58, 6 April 2023 (UTC) github/orcca-uwo/BPAS/ was most recently released in 2019 and has only 10 github stars (fewer than the number of listed authors) and 1 issue in the bug tracker, the homepage bpaslib.org has not been updated in 2 years and has an expired SSL certificate. If this is part of Maple, it’s not at all clear from looking at the project page. Why do you think this bug tracker issue is an acceptable source under WP:RS? I would recommend removing the text "is used in the Basic Polynomial Algebra Subprograms (BPAS) open source library." from the article, which seems to be self-promotion of some kind? But that should be discussed at the relevant talk page rather than here, where it is a distracting off-topic digression. –jacobolus (t) 21:32, 6 April 2023 (UTC) It cannot have expired SSL sertificate. SSL does not exist anymore. TLS 1.2 is used there. Also, I suppose you use a time machine. Because the certificate only expires May 30 2023. Valery Zapolodov (talk) 03:19, 7 April 2023 (UTC) Hm, strange. I guess I didn’t check carefully. When I navigated to that site in my browser it initially displayed the scary “invalid certificate” page which I had to click through a couple disclaimer pages to dismiss. Typically this happens for an expired cert, but maybe something else was wrong with it. Perhaps it was a temporary glitch. Shrug. Edit: a couple of 'certificate checker' sites claim: "Certificate chain is incomplete, missing intermediate(s)". This remains all off topic here though. Do you mind if I collapse these off-topic comments? –jacobolus (t) 07:04, 7 April 2023 (UTC) And BTW Covanov name is mentioned in https://en.wikipedia.org/wiki/Multiplication_algorithm So the whole "student" thing is IMHO wrong. Collapse them, sure. I agree that https://www.hardenize.com/report/bpaslib.org/1680885564#www_certs shows it has only one cert instead of at least 2. Still, that is a new thing, since it allows to decrease load on the network and everyone has those 2 out of 3 certificates anyway. Valery Zapolodov (talk) 16:36, 7 April 2023 (UTC)

If sources are questioned here, whataboutism pointing to their use in Liouville number or another article is not really germane unless part of a broad Wikipedia consensus. Wikipedia is made by thousands of volunteers around the world, and there are many pages which unfortunately contain questionable sources (some much more questionable than these). These are constantly being removed, but there are always more of them. The question here should instead be: do these specific sources meet the criteria at WP:RS, or if not, are they important enough that we should WP:IGNORE the usual guideline? Personally I think the Alekseyev paper should pretty clearly qualify for inclusion under WP:USEBYOTHERS and WP:SELFPUB (“Self-published expert sources may be considered reliable when produced by an established subject-matter expert, whose work in the relevant field has previously been published by reliable, independent publications.”) The second paper seems less clear cut. Would user:MrOllie and user:JBW, who removed these from the article, care to comment? –jacobolus (t) 17:48, 7 April 2023 (UTC)

Ideally someone writes to Alekseyev and requests him to publish the paper in some big journal. This result is already discussed in full in many other papers. Valery Zapolodov (talk) 19:03, 7 April 2023 (UTC)

If it is being used by others, can one of those others be cited as well? It is always better to include a secondary citation when available. That would remove any objections I have. MrOllie (talk) 23:24, 8 April 2023 (UTC)

Here is one, Chen, Sully F.; Pearse, Erin P. J. (2020). "The irrationality measure of π as seen through the eyes of cos(n)". Elemente der Mathematik. 75 (4): 152–165. arXiv:1807.02955. doi:10.4171/EM/417. This one looks it came out of a student project, but I haven't looked too closely. –jacobolus (t) 23:35, 8 April 2023 (UTC)

References

the long paragraph about Zeno's paradox seems a bit out of scope for the lead

I realize this is trying to help non-technical readers, but it feels like too much for the lead section, and this lead section currently seems way too long. I would recommend cutting it down to a couple sentences or moving it wholesale to somewhere else in the article (it seems like it might fit better in the 'history' section alongside discussion of the method of exhaustion and the like, or maybe a new section could be added immediately after the lead about this philosophical/definitional question). –jacobolus (t) 18:20, 16 April 2023 (UTC)