Talk:1 − 2 + 3 − 4 + ⋯

As long as we're talking nonstandard summation methods ...

The last sentence of the article reads:

" For example, the counterpart of 1 − 2 + 3 − 4 + ... in the zeta function is the non-alternating series 1 + 2 + 3 + 4 + ..., which has deep applications in modern physics but requires much stronger methods to sum."

But wait! If S = 1 + 2 + 3 + 4 + ... then

2S = 2 + 4 + 6 + 8+... and

-3S = S - 2(2S) = 1 + 2 + 3 + 4 + ... - 2(2 + 4 + 6 + 8+...) = 1 - 2 + 3 - 4 +....

So if the last summation has already been shown to have a nonstandard summation method assigning it a value of 1/4, then the sum S = 1 + 2 + 3 + 4 + ... can be readily assigned the value (-1/3)(1/4) = -1/12 without requiring "much stronger methods to sum".Daqu (talk) 13:24, 22 October 2015 (UTC)

Does all of this equal 0 or something i don't get it Porygon-Z 23:16, 13 April 2019 (UTC) — Preceding unsigned comment added by Porygon-Z474 (talk • contribs)

Basically, the series is not "summable" in the normal sense because adding more and more terms does not make the series settle down on any value. But it is possible to generalise the notion of summation to cover such series, and when you do that you find that the only reasonable answer is 1⁄4. Double sharp (talk) 13:11, 1 March 2020 (UTC)

Knopp p.491

Does anybody know what resource exactly is cited here? Special:Diff/112109517. Nothing of Knopp's was added in the References section, and the reference just says "Knopp p.491" without a corresponding item in the Bibliography for 13 years now. (The original research phrasing "there appears" has been deleted in the meantime.) @Melchoir: ping as the author, although I see you as inactive... Gikü (talk) 20:45, 22 April 2020 (UTC)

Ah yes, that would be Theory and Application of Infinite Series, LCC 74-166185. Melchoir (talk) 08:50, 18 August 2020 (UTC)

WP:URFA/2020

I'm looking over this as part of the ongoing FA sweeps. There's a couple areas here that need some work here against the featured article criteria. Several places lack citations, the Weidlich source seems to fail WP:SCHOLARSHIP as it is only a masters thesis and does not seem to be widely cited, and there are tone issues, such as two places in the explanation of the formula where the reader is directly addressed as "we". This should be fixable if someone has the time and knowledge of the topic, although since big chunks of this article went straight over my head, that person is not me. Hog Farm _Talk 21:37, 21 March 2021 (UTC)

I removed the two instances of first person (outside a direct quote). Replacing the Weidlich source for Borel summability will require more effort — it is not easy to search for this specific series, and some sources that discuss both Borel summability and this series (like Osler's "Can divergent series be of value") skip over the detailed calculation for the Borel sum of this specific series because its Borel summability is an immediate consequence of its Abel summability. As for claims of missing sources, more specifics would be helpful, so we can tell whether the claims in question need specific sourcing or are more in the way of general knowledge that people familiar with this area would reasonably be expected to know. —David Eppstein (talk) 07:34, 2 June 2021 (UTC)

I gave up on finding someone else who does this precise calculation and instead found a better source for the general statement that Euler summability implies Borel summability. @Hog Farm:: You wrote "Several places lack citations". Can you please be more specific? —David Eppstein (talk) 06:54, 9 June 2021 (UTC)

Will try to look through soon. I'm busier right now with starting a new job. Hog Farm _Talk 10:36, 9 June 2021 (UTC)

@David Eppstein: - It looks like I'm gonna have to push looking at this off until tomorrow (taking a look at Euler for the FAR took a bit of time), but I'm promise I'll get to this. If I haven't posted anything in a couple days, please ping me again. Hog Farm _Talk 01:48, 10 June 2021 (UTC)

Okay, so reading through this again most of this is probably self-proving. I'm not familiar with this subject, so there's a few areas that I'm not sure if it's obvious or not, so please be forgiving if I say anything stupid, because I really don't understand a lot of this subject.

• "Since all forms of Cesàro's theorem are linear and stable" - is it common knowledge that all forms of the theorem are linear and stable?
• "but here one needs the stronger Lagrange form of Taylor's theorem." - is the need for the stronger form common knowledge?
• and is "which has deep applications in modern physics but requires much stronger methods to sum." common knowledge?

I'll be removing this article from the noticed list, as I don't think the issues are anywhere close to a FAR, upon second reading. I think a copy edit would be nice here, as phrasings such as The last convergence sum is the reason illustrate why negative even values of Riemann zeta function are zero are a bit clunky, and someone familiar with math might want to check to make sure all the math markup is formatted in the preferred style, but this shouldn't be a giant project from this point. Hog Farm _Talk 01:09, 11 June 2021 (UTC)

Thanks! Despite the removal from the noticed list I'll try to find time soon to address those three points, if nobody else gets to them first (they all look like the sort of thing that require a little time and effort rather than just immediate edits). —David Eppstein (talk) 01:56, 11 June 2021 (UTC)

Error on Generalization Part

I think it's exactly the same argument as the talk with the title "Error?", though it stopped since 2007 so I've decided to make a new one.

In the Generalization Part of this article, the very first equation is the following:

$$1-2^{n}+3^{n}-\cdots ={\frac {2^{n+1}-1}{n+1}}B_{n}$$

And right after the equation, it says that for positive even integer n, this equation reduces to 0. However, the Bernoulli numbers B_n should have the value 0 at odd integers bigger than 1.

Searching for the other equations related to this one, and found that the left side of the equation can be represented using the Dirichlet eta function. According to the definition, the left side becomes $\eta (-n)$, and equation becomes the following:

$$\eta (-n)={\frac {2^{n+1}-1}{n+1}}B_{n}$$

Also, in the Dirichlet eta function article, on the Particular values section, there is a following equation:

$$\eta (1-k)={\frac {2^{k}-1}{k}}B_{k}$$

If we substitute k with n+1, we get the exact equation above, except the subscript of B becomes n+1. And this equation is consistent with the fact that this equation becomes zero at even integers, since Bernoulli numbers are 0 at odd integers bigger than 1 where n+1 is odd if n is even.

Therefore, it seems like the equation should be the following, instead of the current one:

$$1-2^{n}+3^{n}-\cdots ={\frac {2^{n+1}-1}{n+1}}B_{n+1}$$

Please let me know if there's any error about this new equation. ---- Meaningful0309 (talk) 14:58, 30 August 2021 (UTC)

The Hardy reference asserts that $1-2^{2k+1}+3^{2k+1}-\dots =(-1)^{k}{\frac {2^{2k+2}-1}{2k+1}}B_{k+1}$. This is consistent with the present article only if (1) one observes the convention (mentioned at the Bernoulli number#Notation) of writing $B_{n}$ in place of $B_{2n}$ and (2) Meaningful0309 is correct. Therefore I have changed the article. (I was not able to quickly access the relevant page of the second citation, to Knopp.) --JBL (talk) 00:16, 21 January 2023 (UTC)

Weidlich, pp. 52–55.

Cannot find the full citation for footnote Weidlich, pp. 52–55 in oldid 1180441176 while I was trying to improve the citation and references, after which I decided to put the work on hold. Dedhert.Jr (talk) 01:13, 17 October 2023 (UTC)

The reference was removed by David Eppstein in this edit. --JBL (talk) 19:55, 17 October 2023 (UTC)

I suspect my rationale was a combination of the reference being unnecessary and, in general, master's theses being only borderline-reliable. —David Eppstein (talk) 20:05, 17 October 2023 (UTC)

Will put them back, and also change to sfn (or probably harv)... Dedhert.Jr (talk) 02:42, 19 October 2023 (UTC)

It hardly seems necessary but whatever. At least you didn't also put back the boring calculation showing that in this case (like in every case) the Borel sum exists and equals the Euler sum. —David Eppstein (talk) 06:29, 19 October 2023 (UTC)

Yeah. I was trying to fix the references and citation style, anyway; boring calculation is not my thing. Dedhert.Jr (talk) 09:51, 19 October 2023 (UTC)