Talk:Binomial theorem

Please clarify statement in opening paragraph

"When an exponent is zero, the corresponding power expression is usually omitted from the term." I was thoroughly confused by the intended meaning of this. Does it simply mean that any x or y raised to the power 0 is taken to be 1? If so, why not say exactly that? Otherwise, it's unclear what exponent is being referred to, what power expression is being referred to, and what is meant by "associated" power expression. If that is *not* what is meant, then I would request that the statement be rewritten to clarify the intended meaning. Craniator (talk) 17:36, 14 June 2015 (UTC)

 Done Bill Cherowitzo (talk) 04:52, 15 June 2015 (UTC)

Bill Cherowitzo's version was much better than the previous one, but I think consideration of weight dictates that we should just ignore this issue in the lead paragraph. It's a minor technical detail, and it was receiving more prominence than binomial coefficients. We can address it in the body, instead. --JBL (talk) 15:37, 18 June 2015 (UTC)

Good point. Should have thought of doing that myself. Thanks. Bill Cherowitzo (talk) 17:06, 18 June 2015 (UTC)

Reference the "Binomial series" and remove examples when n is not a positive integer.

I suggest that following the first sentence: "In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial." that a sentence is added stating "For binomial expansions where the exponent is not a positive integer, see the article regarding the Binomial series."

Then, I suggest that the examples (pasted below) regarding square roots (n=1/2) and negative exponents (n=-1) be removed, because they are clearly out of scope of this article.

"Another useful example is that of the expansion of the following square roots:

${\sqrt {1+x}}=\textstyle 1+{\frac {1}{2}}x-{\frac {1}{8}}x^{2}+{\frac {1}{16}}x^{3}-{\frac {5}{128}}x^{4}+{\frac {7}{256}}x^{5}-\cdots$

${\frac {1}{\sqrt {1+x}}}=\textstyle 1-{\frac {1}{2}}x+{\frac {3}{8}}x^{2}-{\frac {5}{16}}x^{3}+{\frac {35}{128}}x^{4}-{\frac {63}{256}}x^{5}+\cdots$

Sometimes it may be useful to expand negative exponents when $|x|<1$:

$(1+x)^{-1}={\frac {1}{1+x}}=1-x+x^{2}-x^{3}+x^{4}-x^{5}+\cdots$"

See from http://www.haverford.edu/physics/MathAppendices/Binomial_Expansions.pdf, which apparently are public domain and in my opinion has a better "Statement of the theorem" than this article. For example, it also states that "The series in eq. (1) can be used for any value of n, integer or not, but when n is an integer the series terminates or ends after n+1 terms." 207.107.66.194 (talk) 15:00, 6 July 2015 (UTC)

I agree with both proposed edits (removal of out of scope examples and addition of a sentence pointing to the binomial series, possibly to the lead of the article where it can easily be located). Sławomir Biały (talk) 17:38, 6 July 2015 (UTC)

References

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Examples

In the examples section there are some with fractional or negative exponent, but the generalized binomial theorem that deals with those cases is only mentioned later. Up to this point only non-negative integer values of $n$ have been considered and the summation is always from 0 to $n$. — Preceding unsigned comment added by Pacosantosleal (talk • contribs) 12:24, 25 November 2015 (UTC)

Meaning of n choose k

$Does{n \choose k}$ mean n_k ? — Preceding unsigned comment added by Skk146 (talk • contribs) 15:49, 27 November 2016 (UTC)

History

Joel B. Lewis I think it's better to cite Rashed here, because the source used is St Andrews, but while O'Connor and Robertson quote Rashed's statement, and apparently agree with it, the statement itself is Rashed's alone, and would be best cited directly to the source where he makes it. Not doing so when the source is readily available online is very poor scholarly practice. Best regards.---Wikaviani (talk) 00:37, 5 May 2018 (UTC)

recent edits

To help settle this recent conflict, I'm starting this talk page section. To the IP who keeps edit warring their preferred version in, there are a lot of inappropriate spacing changes you're making, as well as things like changing the capitalization of piped links (the link part, not the visible part); this should not be done. There are also a lot of invisible wikicode changes you're making. These are fine in and of themselves, but edits should not be made just to do that. See WP:COSMETICBOT for further info about these. WP:BRD is a pretty good idea. If someone reverts your changes from the status quo, it's really up to you to try to justify the changes, not make snarky retorts that don't address the disagreement. So again, please discuss here rather than trying to re-add. –Deacon Vorbis (carbon • videos) 21:05, 28 April 2019 (UTC)

The only reply to your only point, which is to assert "inappropriate", is to assert "appropriate". I am not going to further engage in pointless rhetoric with people want to claim that some standard should be met, never stating that standard, and moving it when it pleases them.80.65.247.112 (talk) 22:57, 28 April 2019 (UTC)

If you're unwilling to discuss, then you have no basis to continue to insist on your changes. If there's objection to those changes, you need to pause and resolve that objection rather than just plowing ahead. I've made some specific points here which should be addressed. Since you're unhappy with my use of "inappropriate", I'll elaborate on the spacing. You're using little-used manual spacing templates that aren't well known, aren't needed, produce inconsistent results, run contrary to MOS:MATH, and thus make maintenance more difficult. You've also added delimiter sizing commands that aren't needed – like $(x^{n})$ vs. $\left(x^{n}\right).$ This produces no difference in rendered output. A small handful of the changes you made were reasonable, mainly display mode vs. text mode for binomial coefficients. I reinstated those (which you took back out in your rush to revert me), but the vast majority were either unnecessary or actively harmful. I don't know why you feel such a need to see these edits stand, but there are good reasons why they shouldn't. –Deacon Vorbis (carbon • videos) 23:18, 28 April 2019 (UTC)

I notice that 80.65.247.112 doesn't want to say much on this Talk page, and doesn't leave edit summaries to give some hint as to his motivation. Neither of these habits is compatible with the way we do things on Wikipedia. Conversely, Deacon Vorbis not only shows a willingness to use the Talk page constructively, he shows an enthusiasm for doing so. Therefore I endorse the standard advocated by Deacon Vorbis. The editor at 80.65.247.112 should either adopt the Wikipedia way of doing things, or get used to the idea that all his hard work will one day amount to nothing. It should be an easy decision. Dolphin (t) 13:16, 29 April 2019 (UTC)

Multiple issues

^{NOTE: The version of the page being discussed}

Currently only a history is provided with enough citations and the rest is poorly sourced. Would be nice of someone adds a couple of links out there. DAVRONOVA.A. ✉ ⚑ 08:27, 17 June 2019 (UTC)

That's only one issue :p. Abstractly you are certainly right; concretely, are there any particular places you think are suspect (either, might be wrong, or with inappropriate weight, or original research)? Because "please make the article better" alone is not very constructive. --JBL (talk) 14:28, 17 June 2019 (UTC)

Original Research in substitution of of e^ax and e^bx in general Leibniz formula section of generalizations

Hi! The substitution of e^ax and e^bx into the general Leibniz formula seems to have no sources and thus seems to be original research. Although I understand that this substitution is correct mathematically, but there are no published sources having the same idea. Please take a moment to review my edit and verify my changes. The two changes I made are:

• Added [original research?] tag to the e^ax, e^bx substitution claim
• Added [verification needed] to the source "Calculus in One and Several Variables" by Robert Seeley since the book does not contain the Leibniz formula at all. The "Calculus for One Variable" can be found here, available for digital borrowing. This is single-variable calculus part of the combined book. Therefore, the Leibniz formula is expected to be in here. However, after extensive searching, no record of the Leibniz formula was found here.

I understand that a similar issue was addressed under the "Multiple issues" section, but I wanted to keep a concrete discussion just for this change. If a citation is found, please feel free to enter a citation. Otherwise I will be forced to remove the content according to the Wikipedia: Original research policy. Thank you. --Dh*Phoenix (talk) 17:40, 20 June 2020 (UTC)

This isn't OR; I'm having trouble finding a good ref for it, but a quick google search finds a lot of message-board type expositions of this fact. While a source would be ideal, the ease with which one finds non-RS discussion + WP:CALC probably lets this one stand. As for the Leibniz rule, there's another source at the main article that you could probably use if you think it's better. You could always just cite Abramowitz & Stegun too, or probably hundreds of other Calculus textbooks. –Deacon Vorbis (carbon • videos) 18:54, 20 June 2020 (UTC)

Ref added for the binomial theorem as a corollary of the Leibniz rule. –Deacon Vorbis (carbon • videos) 19:38, 20 June 2020 (UTC)

Yes, I'm sorry: this is not original research. Thank you for the source! And yes, I find the main article Leibniz rule citation better: I'll correct that.--Dh*Phoenix (talk) 05:47, 21 June 2020 (UTC)

Applications

The section on the infinite series for e is alright up until "This indicates that e can be written as a series:", at which point rigor and accuracy is lost. See Rudin, Priniciples of Mathematical Analysis, Section 3.3.1, for a proof. The mistake in the page as it exists is that it is not sufficient to say that the in the limit kth term is 1/k! implies that the entire series of 1/k! terms is equal to the original series. — Preceding unsigned comment added by 192.80.55.86 (talk) 18:20, 29 November 2021 (UTC)