Talk:Basel problem

Headline text

I'm thinking of renaming this article "The Basel Problem" and adding a bunch of historical remarks and putting in broader context. I still think the proof can stay here, it would just be a part (about half) of the whole article. I don't think it would make it too long, and people not interested in the proof could just skip that section. I thought of having 2 articles, but it seemed redundant and just giving the proof without any remarks about the problem seemed strange. Revolver 15:31, 26 Feb 2004 (UTC)

I manually renamed this article Basel problem, (since the system wouldn't let me do this directly) so this article should be deleted. Revolver 20:28, 26 Feb 2004 (UTC)

Is it not worth keeping as a redirect? Angela. 02:18, Feb 29, 2004 (UTC)

Possibly...I find it hard to imagine someone linking to it, but it can't hurt. Revolver 02:41, 29 Feb 2004 (UTC)

---

That's a good idea...I forget simple things like, "why does this sum converge in the first place", assuming everyone has seen p-series in calculus, or something. There may be some other simple ways to approximate this that people used before Euler that might be interesting. Revolver 22:18, 3 Mar 2004 (UTC)

Overlooking the obvious?

It isn't clear to me from the article why this problem is called the Basel problem. Dataphile 19:49, Aug 22, 2004 (UTC)

Basel is the name of a town, somehow the town is related to the origin of the problem. I don't know exactly -- in math, so many things have so many names, math people often don't know why something's called what it is. Revolver 17:51, 24 Aug 2004 (UTC)

The Bernoulli family, which worked on the problem for a long time, were located in Basel. Johann Bernoulli taught Euler at the University of Basel.

Picture as proof

"First note that 0 < sin x < x < tan x. This can be seen by considering the following picture:"

It's definitely a cool picture. Can its owner change the 'theta' to an x? Do we need to point out that the angle is equal to the length of the circular arc AD?

I can 'see' that sin x < x, but I can only convince myself that x < tan x using calculus (e.g., the derivative of (tan x - x) is tan²x, which is > 0 for x in (0, pi/2), ...). Is that a failure of imagination on my part? Perhaps it's obvious that tan x grows faster than x.

Buster79

The area of the triangle OAE is tan(θ)/2. The area of the circle sector OAD is θ/2. The circle sector is contained within the triangle, so θ < tan(θ). Fredrik Johansson - talk - contribs 18:18, 24 January 2006 (UTC)

Thanks. I've added this to the article. Buster79 23:08, 2 September 2006 (UTC)

You can prove it easily by using Fourier transform

I will not go into details, but the proof starts with the function absolute value of x on the interval [-pi,+pi] You calculate the Fourier transform of the function. Later you can seperate the sum into odd and even numbers and find out that the sum equals pi^2/6.

The Riemann zeta formula can be approached by Parseval's theorem.

The point of the proof in this article is that it needs only elementary methods. There are several proofs of this problem (there is an article called "Six ways to sum a series" which gives 6 of them), but most of them require more advanced machinery.

Why is the same Fourier series proof given twice?

It looks ridiculous. The two proofs are clearly entirely the same but one is stated in a pretentious way. Someone should fix this. The only reason I didn’t delete it immediately is because I’m unaware of Wikipedia’s protocols. — Preceding unsigned comment added by 2600:1010:B002:D03B:F5FB:3BB3:90A5:C78C (talk) 07:10, 18 February 2020 (UTC)

Basel problem extended

The Basel problem can be extended to find the closed forms for every N. An approximate sequence can be found in the OEIS, A111510. Included is an expression of Pi where the odd and even terms of Triangular(n)define the differences. Would the contributers to the Basel problem pages care to comment and to suggest the best way to include this? Marc M. 20-6-06

Irrelevant topic

The last topic in the article on the Basel problem reads as follows:

"Use in the calculation of π: In 1881, Ernesto Cesaro showed that the probability of two integers being relatively prime equals 6 / π2, which is the reciprocal of ζ(2). By the above proof, Cesaro's theorem thus allows a value for π to be calculated from a large collection of random integers, by determining the proportion of them which are relatively prime."

I don't see how this uses the result of the Basel problem, ζ(2) = π² / 6, at all; it just mentions it in passing. Likewise, the phrase "By the above proof" is unnecessary, since the application of Cesaro's theorem goes along just fine without using the above proof at all. Moreover, it's very misleading to say that π can be calculated by this method. If you pick 100 pairs of random numbers from 1 to 10, and every possible pair shows up once, you will "calculate" π ≈ 3.086. (There are 63 pairs that are relatively prime.) No matter how many pairs are selected, the result will be an algebraic number, and π is transcendental. I suggest the entire paragraph be removed from the article. Did I miss anything? Gwil 20:03, 27 September 2006 (UTC)

Fundamental Theorem of Algebra unnecessary

I object to the following:

If p(t) is a polynomial of degree m, then p has no more than m distinct roots. Proof: This is a consequence of the fundamental theorem of algebra.

The statement "If p(t) has degree m, then p has no more than m distinct roots" can be proven by elementary methods, whereas the FTA is much more difficult (it's usually proven with complex analysis). Specifically, the "no more than m distinct roots" theorem follows from the factor theorem ( (x - r)|p(x) if and only if p(r) = 0, a consequence of polynomial division), together with an induction argument to show that the product of (x - r_j) over all the roots r_j of p(x) divides p(x), and an application of the fact that degree(f * g) = degree(f) + degree(g) for nonzero polynomials f and g.

Moreover, the "no more than m distinct roots" theorem holds in the polynomial ring of any integral domain, whereas the FTA is a very special fact about the complex numbers.

I would recommend that the line "Proof: This is a consequence of the fundamental theorem of algebra." be replaced by an outline of the actual proof from the factor theorem. --Gene496 08:35, 27 July 2007 (UTC)

While Euler is credited with the solution, i.e., an exact answer, is pi^2/6 an exact number? pi itself is the sum of an infinite series as is the original Basel series! SEIBasaurus 70.118.127.94 17:41, 7 August 2007 (UTC)

What you need to know

Just a comment on the "What you need to know" sub-section... love this, extremely helpful, should be present in more articles! Error792 03:20, 6 August 2007 (UTC)

Coprime?

Would it be worthwhile to add a wikilink to coprime, regarding the application of ζ(2) in evaluating the probability that two randomly chosen numbers are coprime? I just added a wikilink in the other direction, from that article to here. —David Eppstein (talk) 22:11, 15 December 2007 (UTC)

Long proof

This article has bothered me for a long time. I have looked for elementary proofs that were also simple, but found none. Perhaps it is not surprising that it took Euler quite a while to figure it out.

I like the elementary proof precisely because it is elementary. However, I think it is also very complicated and in most cases will not be interesting to the reader. On the one hand, I think it is too lengthy to be readable by most laypersons. On the other hand, I would think that almost anyone who is familiar with Fourier series would prefer that proof, because one can understand it at a glance, and remember it forever more (whereas I could not reproduce this proof without some effort, and I do math research...)

Because this proof is so well written and also elementary, I think it should be moved to its own article. I like Euler's "proof", and it should stay, and for completeness, we could leave the Fourier Series argument which I just put in, because that would give a very short airtight argument. However, right now, I think that most of our readers must stop reading somewhere in the "What you need to know" section.

Any objections?

Loisel (talk) 19:55, 23 July 2008 (UTC)

I think that several proofs on the same page is a good idea. The Fourier series argument is a very welcome addition, it's the proof usually taught in calculus courses. There are more proofs given in the external links. I prefer to have good outlines or section titles for the proofs instead of distributing them over several articles. So the reader can make a choice according to his/her level of mathematical education or specific interest, compare with Proof that the sum of the reciprocals of the primes diverges and Inequality of arithmetic and geometric means. Schmock (talk) 00:37, 24 July 2008 (UTC)

Nono, what I'm proposing is to segregate most of the proofs to an article, maybe, Proof that zeta(2)=π^2/6, much like Proof that the sum of the reciprocals of the primes diverges. This article could be more about the history of the Basel problem and maybe one proof as short as possible.

No?

Loisel (talk) 02:18, 24 July 2008 (UTC)

I prefer to have the elementary proof on the same page as the history of the problem, because this proof is potentially readable for people without a university education in mathematics. When we have several (at least 4) advanced proofs, a separate article might be a good idea. Personally, I prefer to have some proofs in the article zeta constant, in particular for ζ(2n). As far as I remember, Fourier methods also work in this more general case, but other proofs might be shorter. Schmock (talk) 08:06, 24 July 2008 (UTC)

My understanding of zeta(2n) is that you find an nth antiderivative of f(x)=x. So for instance, assume that $g^{(n)}(x)=x$. So g is periodic and n times differentiable. The constant terms are prescribed by the homology or something. Clearly, g(x) is a polynomial and for a fixed n, you can compute it. Hence, you can compute \int g^2. Furthermore, ${\hat {g}}(k)={1 \over (ik)^{n}}{\hat {f}}(k)$ and that gives you the necessary formula.

But I don't know if there's an explicit formula for g. Do you know?

Loisel (talk) 23:44, 24 July 2008 (UTC)

A slicker proof from Fourier series

What is meant with "worked out in that article"? If it's the last link than the link is dead. Furthermore does the Parseval's identity on wikipedia not give the desired result for this prove. Mostly because here the Parseval's identity is stated wrong. 86.95.192.7 (talk) 08:59, 8 November 2008 (UTC)

Problem in proof.

It seems to me that the proof shown in the section "Euler attacks the problem" has a fallacy. Before I explain it, it should be said that the proof naturally can be used after it has been modified a bit.

It is stated that:

${\begin{aligned}{\frac {\sin(x)}{x}}&{}=\left(1-{\frac {x}{\pi }}\right)\left(1+{\frac {x}{\pi }}\right)\left(1-{\frac {x}{2\pi }}\right)\left(1+{\frac {x}{2\pi }}\right)\left(1-{\frac {x}{3\pi }}\right)\left(1+{\frac {x}{3\pi }}\right)\cdots \\\end{aligned}}$

Actually this is only true for $x=n\cdot \pi$ where $n=\pm 1,\pm 2,\pm 3,\dots \,$ since both sides equals 0 (which has been mentioned earlier in the proof). Generally we have:

${\begin{aligned}{\frac {\sin(x)}{x}}&{}=a\cdot \left(1-{\frac {x}{\pi }}\right)\left(1+{\frac {x}{\pi }}\right)\left(1-{\frac {x}{2\pi }}\right)\left(1+{\frac {x}{2\pi }}\right)\left(1-{\frac {x}{3\pi }}\right)\left(1+{\frac {x}{3\pi }}\right)\cdots \\\end{aligned}}$

But as noted before the proof is still valid. It should just be noted that the following calculations in the proof only are valid for $x=n\cdot \pi$ where $n=\pm 1,\pm 2,\pm 3,\dots \,.$.

I haven't changed the article since I want to be sure I'm not making a mistake, but if you agree then please correct it.

—Preceding unsigned comment added by 82.211.210.23 (talk) 14:34, 15 November 2008 (UTC)

You are making a mistake. The formula is correct with a=1. The only problem is that Euler had no proof of this fact; it was compelling because it worked to solve the problem, but Euler died before the tools to prove it were developed. Nowadays this product expansion can be derived from Weierstrass's factorization theorem in complex analysis. 97.127.142.123 (talk) 06:36, 16 October 2009 (UTC)

Flaw in Euler's method

The fact that sin(x)/x has zeroes at +/-(pi, 2pi, 3pi, ...) doesn't mean sin(x)/x can be expressed as the product of terms in the form (1 - x/pi).

Maybe I'm missing something here but when you express a finite polynomial in terms of its linear factors (which is the assumption Euler is making here and the basis of his proof), it is of the form: (x - root1)(x-root2)...

x-pi is not the same as 1-x/pi —Preceding unsigned comment added by 76.182.194.195 (talk) 15:01, 19 February 2010 (UTC)

For a polynomial of degree n with roots z₁, z₂, ..., z_n; p(z) = a (z-z₁) (z-z₂) ... (z-z_n) -- that is, the polynomial is determined by its roots only up to a constant multiple. When 0 is not a root we can equally write p(z) = b (1-z/z₁) (1-z/z₂) ... (1-z/z_n) where b = a (-z₁) (-z₂) ... (-z_n). This form has its advantages, one being that b = p(0). -- ToE^T 16:45, 4 July 2010 (UTC)

Proof through Complex Analysis

Isn't there a proof which uses complex analysis for the same? --Hirak 99 (talk) 08:53, 12 August 2010 (UTC)

There's a proof using complex analysis already in the article, as the first proof: Basel_problem#A_rigorous_proof_using_Fourier_series. For many other proofs, see Robin Chapman's collection, also linked in the article. Shreevatsa (talk) 09:01, 12 August 2010 (UTC)

wouldt it be...

would it be too dare if someone (including me) could add the "folklore" [1]? it is about some other sum of reciprocals' solved problem :)

--kmath (talk) 02:33, 26 September 2012 (UTC)

broken link

Euler's solution of the Basel problem – the longer story PDF (61.7 KB) — Preceding unsigned comment added by 97.86.236.127 (talk) 20:51, 12 June 2013 (UTC)

the record holder's rectangles area will always be greater than the ideal rectangle - citation needed

"Because π is a transcendental number, the record holder's rectangles area will always be greater than the ideal rectangle."

What's the source for this? I don't see how π being transcendental, i.e. not a solution to an algebraic equation - implies that a well defined packing algorithm has to be greater than the ideal rectangle. For instance is easy to write out a program that can print out all the digits of π one after another. That a number is transcendental doesn't normally have much by way of implications for computability. Am going to add a citation needed tag to this. If it is "obvious" for some reason do explain and add some hint to the text to help readers who don't get it. Thanks! Robert Walker (talk) 18:26, 7 September 2014 (UTC)

A transcendental number has the implication of not being able to be computable exactly only approximately. Because of this, "the record holder's rectangles area will always be greater than the ideal rectangle". Even if you find pi to 10^1000 digits it is only approximately correct (the 10^1000 + n digit number would yet to be found). 98.95.5.178 (talk) 02:59, 11 September 2014 (UTC)

This is as confused as the sentence in the article. Either we have a proof that the rectangle with the exact area can be packed or we don't. If we do have such a proof, then there is a packing that is NOT greater than the ideal area. If we don't have such a proof, then any packing we do have uses greater area. This would be true regardless of whether the number in question were rational, algebraic, or transcendental. And whether or not such a packing exists is also different than whether the packing is computable. Additionally, transcendental numbers such as π are no more computable or uncomputable than rational numbers like 1/3 — neither is exactly representable in binary but both can be computed easily to arbitrarily high precision. In short, I agree with Robertinventor that being transcendental doesn't have much to do with the inequality between known and ideal packings. I think we should just remove this sentence. —David Eppstein (talk) 03:19, 11 September 2014 (UTC)

Removed square packing section

I removed the section on square packing. It was poorly written (see above comments) and had nothing to do with the subject of the article anyway.24.144.122.240 (talk) 19:22, 14 October 2015 (UTC)

Formatting per WP:MSM and WP:MOSMATH#Multi-letter names

@Slawekb: I see in your second revert message a correct point, and will fix the italicized use of trig functions. However, that was a secondary point of the edit, and your revert "throws out the baby with the bath water" by reverting instead of editing and consulting. Do you object to the entire edit or just the use of itallics for the trig functions? —Boruch Baum (talk) 13:08, 15 December 2015 (UTC)

The edit that I reverted replaced \sin with {{sin}}. Yes, I object to this edit in its entirety. Sławomir
Biały 13:19, 15 December 2015 (UTC)

@Slawekb: And would you care to share with anyone why may that be? —Boruch Baum (talk) 13:23, 15 December 2015 (UTC)

It is a clear and unambiguous violation of our manual of style, and basically all mathematics typesetting conventions in existence. Trigonometric functions are always typeset in roman case. But really the point of WP:BRD is that the onus is on you to justify the edit. I've already given my answer (WP:MSM). You need to say why ignoring the manual of style is justified. Sławomir
Biały 13:31, 15 December 2015 (UTC)

@Slawekb: You don't seem to be paying much attention either to what you are doing or what I have been saying, so let me try again. My edit primarily was a fix to excess vertical whitespace on the page (as was explicitly documented in the edit description, which you apparently ignored). You reverted a lot of work because you found a minor quibble with itallics that I have already agreed (above) to fix (maybe you didn't read the first sentence of the first paragraph of this thread). Before I continue to edit the page to comply with your correct point about the use of itallics, let me know what other objections you have with the edit, so that the main point of the edit can remain. As a general matter, if you re-read and think about what you've written, I think you'll come to agree that its neither sufficient nor courteous to cite a huge multi-faceted page as a source of objection without a specific description of your specific objection. —Boruch Baum (talk) 13:39, 15 December 2015 (UTC)

@Slawekb: Also, please use the {{replyto}} when adding to this thread, so that I receive notification of your new addition. —Boruch Baum (talk) 13:46, 15 December 2015 (UTC)

Ok, which parts of this edit do you wish to discuss? Please be specific. The only thing I see in that diff is replacing \sin and \cos with {{sin}} and {{cos}}. Also, I've pointed you to the specific paragraph WP:MOSMATH#Multi-letter functions. This is the entire reason that I object to each and every part of the edit that I reverted. I don't see any other part of that edit that I would even need another more nuanced explanation. But, even if so, in making large-scale typographical changes, the onus is ordinarily on the editor proposing the change to get consensus first. Per WP:MSM:

Large scale formatting changes to an article or group of articles are likely to be controversial. One should not change formatting boldly from LaTeX to HTML, nor from non-LaTeX to LaTeX without a clear improvement. Proposed changes should generally be discussed on the talk page of the article before implementation. If there will be no positive response, or if planned changes affect more than one article, consider notifying an appropriate Wikiproject, such as WP:WikiProject Mathematics for mathematical articles.

--Sławomir
Biały 13:49, 15 December 2015 (UTC)

@Slawekb: Now we're getting somewhere. Great. Take a look at the consequences of the change and you'll see that what it accomplishes is that it GREATLY tightens the vertical white space between the numerator of the fraction and the fraction horizontal line. That is the objective of the edit, and that is what was meant by the edit comment "reduce numerator vertical spacing". If you take a look at the version before I started my edits, you'll see that there is problem with the vertical spacing of all fractions in which a trig function appears, compared to a simple fraction.

The issue of italics is something that I agree with you on, and am prepared to commit to fix, but do you really want that huge vertical and inconsistent whitespace between the types of fractions? —Boruch Baum (talk) 14:13, 15 December 2015 (UTC)

I am not seeing the vertical whitespace problems you're having. That suggests that perhaps this is something best fixed by checking your preferences and browser configuartaion, rather than make changes that go against MoS. It isn't just the italics, but the display of the inline Greek letters. Please use either math tags or the template Template:Math. Sławomir
Biały 14:42, 15 December 2015 (UTC)

@Slawekb: Well, if you're telling me that you don't have the problem, I'm willing to take that on good faith, deduce from your report that my rendering engine is the issue (though that would be strange as its a standard and up-to-date version of Firefox), and remove the objection to reverting the edits I've made. —Boruch Baum (talk) 15:08, 15 December 2015 (UTC)

Basilea Problem

Please add a link to this from "Basilea Problem". John G Hasler (talk) 18:53, 25 February 2018 (UTC)

Google scholar finds zero hits for that phrase, and Google books finds no credible hits. Are you sure it's a standard variant of the name? —David Eppstein (talk) 19:10, 25 February 2018 (UTC)

Basilea is the Italian name for Basel, apparently. –Deacon Vorbis (carbon • videos) 19:11, 25 February 2018 (UTC)

Ok, but "problem" is not the Italian word for "problem", Basel is not in the Italian part of Switzerland, and this is not the Italian version of Wikipedia (where the title of the corresponding article is "Problema di Basilea"). —David Eppstein (talk) 19:55, 25 February 2018 (UTC)

Recent addition

@Contribute.Math: The primary problem with adding this proof is that there's no source for it. It's a cute approach, and one I hadn't actually run across before. But there are already a few proofs in here, and we could add at least 10 more. I'm not sure if it really benefits the article to just catalog proof after proof. On a side note, if you're interested in contributing more to math articles, please also see especially MOS:MATH#TONE (and the rest of the page, too). Constructions like "Let's ..." are problematic. –Deacon Vorbis (carbon • videos) 15:00, 6 December 2019 (UTC)

@Deacon Vorbis: Thank you for your correction. I replaced this time the problematic constructions. I also published the source. It is Euler's first rigorous proof (and second proof) of the theorem. As a result, I believe that it is worth mentionning it in the article. Moreover, it is a relatively short proof. The French wikipedia page refers to it.(see line 13 and "4 La démonstration d'Euler"). Since it is a proof made by Euler, I think that an English source exists. If someone finds it, please replace the French source. December 6 2019 — Preceding unsigned comment added by Contribute.Math (talk • contribs) 16:31, 6 December 2019 (UTC) Contribute.Math (talk) 16:58, 6 December 2019 (UTC)

Please WP:INDENT and WP:SIGN your posts with 4 tildes (~~~~). See H:TP for more information. WP:Wikipedia is not a reliable source. They do in fact give a source, and it's similar to, but different from what you've added. In fact, I found some notes from a talk which credit this proof to a note by Boo Rim Choe in the American Mathematical Monthly in 1987 (see #7). I haven't looked up the original, but if the credit is valid, it wouldn't have the historical significance that you're claiming, and we can probably skip it. –Deacon Vorbis (carbon • videos) 16:49, 6 December 2019 (UTC)

@Deacon Vorbis: The proof has the historical significance that I claim. I found the reference from Euler that proves this fact. As you said, there is a slight difference between Euler proof and the one that I presented. But I chose this proof because it uses the famous and familiar Wallis' integrals.

A single integration by substitution u = sin(t) is needed to connect the two proofs. The second exact Euler proof is currently available on https://lemoid.wordpress.com/ Contribute.Math

One more time, please indent and sign your posts. WP:PINGs will not work if you don't sign your post. I found a source for the version of the proof you're trying to add, and it's not directly due to Euler. There's probably room to add a short discussion of what Euler did; I think the Sandifer source would be good for that. Please slow down and discuss things before continuing to try to re-add material over objections; again, see WP:BRD. –Deacon Vorbis (carbon • videos) 22:27, 6 December 2019 (UTC)

I'm sorry for the technical problems(Identation, signing the posts). I'm new on Wikipedia. I will no longer modify the article before we make a consensus since you insist on this principle. Can you tell me more about the source which shows that the proof was no directly due to Euler ? During my researches, I found many sources that show that the proof was due to Euler. If the proof is not his, then one can just write in the Wikipedia article that he was the first who published it. Here are the sources I mentioned : https://www.apmep.fr/IMG/pdf/Article_probleme_Bale.pdf (pp 17-19), https://fr.wikipedia.org/wiki/Probl%C3%A8me_de_B%C3%A2le#La_d%C3%A9monstration_d'Euler , http://eulerarchive.maa.org/hedi/HEDI-2004-03.pdf (The Sandifer article you mentioned) , https://pdfs.semanticscholar.org/5a8a/5a18e10917d364b61282eb76fd57024bdc0d.pdf How Euler could have used Information & Communication Technology, National institute of Education Singapore (See p.3) Other references are mentionned in the same page. (p.3) Contribute.Math (talk) 10:11, 7 December 2019 (UTC)

Deacon Vorbis, you deleted the last version of the paragraph that I am trying to add, ignoring my last message on this talk page. I find your attitude disrespectful. Indeed, I have been waiting for an answer for more than two weeks on this talk page. As you were not replying, I chose to add the paragraph again, two times. You deleted it two times, without explaining clearly your arguments. Please answer to my messages on this page so that we avoid edit warring. As I said before, please show me the source that you found that shows that the proof is not directly due to Euler. I mentioned four different sources on the article that show that the proof was published by Euler. As I said before, if the proof is not his, then one can just write in the Wikipedia article that he was the first who published it. If you think this is not true neither, one can simply say that he published it. The references that I mentioned in this article prove this fact. Note that in the first paragraph of the article's page, it is written that "it was not until 1741 that he was able to produce a truly rigorous proof". You said that the proof I added was not Euler's proof. If you read carefully the following articles ( https://faculty.math.illinois.edu/~reznick/sandifer.pdf , https://www.apmep.fr/IMG/pdf/Article_probleme_Bale.pdf , https://lemoid.wordpress.com/2014/03/12/basel-problem-arcsin-x-solution/ ), you will notice that the proof I published is exactly Euler's one. I redacted it in a modern style to make it more readable. The modern style enabled me to use modern mathematical rigour. You also said the paragraph is "full of formating issues". Can you tell me what issues you are talking about ? I will try to do something, but you are welcome to fix those issues if you can, since it is a collaborative work.Contribute.Math (talk) 10:25, 22 December 2019 (UTC)

@Contribute.Math: What you've presented was not what was directly due to Euler. Euler's proof also wasn't fully rigorous, because of issues with the interchange of the integrals and sum. It's not up to you to fill in this part; that counts as WP:OR. We also don't need to give the full details here. What we could do is to give a short blurb describing what Euler did at the end of the "Euler's approach" section, sourced to Sandifer. Maybe something like:

In 1741, Euler published a second proof that did not rely on infinite products. In it, he computes the integral

$\int _{0}^{1}{\frac {\sin ^{-1}x}{\sqrt {1-x^{2}}}}\,dx$

by two methods: first directly, and then by expanding the inverse sine as its Taylor series and integrating term-by-term. Equating the two completes the proof.

This is just a basic attempt at something, and it can certainly be tweaked. But it fills in the information without getting bogged down in the details. –Deacon Vorbis (carbon • videos) 15:40, 22 December 2019 (UTC)

This is a good time to remember that we're here to write an encyclopedia, not a textbook. It is often better for our purposes to summarize a proof concisely in prose, rather than to step through it equation-by-equation. Accordingly, Deacon Vorbis's suggestion here is reasonable. XOR'easter (talk) 18:23, 22 December 2019 (UTC)

I agree to delete most details, but not the key details. Otherwise, the reader may not be able to do the proof himself if he wants to. Here is what I would write:

In 1741, Euler published a second proof that did not rely on infinite products. In it, he computes the integral $\int _{0}^{1}{\frac {\arcsin(x)}{\sqrt {1-x^{2}}}}\ dx$ by two methods: first directly, and then by expanding the arcsine as its Taylor series and integrating term-by-term. He then computes $\int _{0}^{1}{\frac {x^{2n+1}}{\sqrt {1-x^{2}}}}\ dx$ with $\displaystyle n\ \in \mathbb {N}$ using an integration by parts. Equating the two gives the value of

$\sum _{n=0}^{+\infty }{\frac {1}{(2n+1)^{2}}}$. He finally decomposes each positive integer into the product of an odd number and a power of two, then uses a geometric series to complete the proof.

Euler's proof also wasn't fully rigorous, because of issues with the interchange of the integrals and sum. The Monotone_convergence_theorem can justify this interchange.

One can also compute $\int _{0}^{1}{\frac {x^{2n+1}}{\sqrt {1-x^{2}}}}\ dx$ using an integration by substitution and recognize Wallis' integrals.

The two last lines may be WP:OR since they contain an original reasoning. However, I think they can be accepted. Indeed, there are two references to other wikipedia pages. Thanks to them, the reader can verify the reasoning with relative ease. (See https://en.wikipedia.org/wiki/Wikipedia_talk:These_are_not_original_research#calculations ) If the two last lines are not accepted, then other paragraphs present in the article such as "A rigorous proof using Fourier series" should not be accepted neither since they contain original reasoning two. --Contribute.Math (talk) 19:31, 22 December 2019 (UTC)

This is still too much, and my previous suggestion still stands. You'll have to understand that I do other things here, and I'd just gotten really tired of discussing this. Among other things, deducing the value of the series from the value of the series with just the odd terms is fairly elementary and doesn't even need to be given in any detail. Adding in one extra little bit to say what the LHS equals, and what the series expansion of the RHS is along with the resulting value of each term would be okay, but that's really about it. I'll try to get to that at some point today, but I'm working on other stuff here too. –Deacon Vorbis (carbon • videos) 18:46, 22 February 2020 (UTC)

Feedback

Noice article 2001:56A:7C52:D300:884B:E679:8F43:9722 (talk) 01:58, 30 December 2021 (UTC)

The 6th note references a product up to n of k, but the expression does not have a k in it.

The 6th note references a product up to n of k, but the expression does not have a k in it. 146.95.224.227 (talk) 18:49, 18 January 2024 (UTC)