Talk:Basel problem

2371:

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 Knowledge 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 (

1886:

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. —

1694:

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. —

84: 74: 53: 1327: 991: 359:

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?

2383:), 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. 22: 355:"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." 1089: 759: 1961:. 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 2990:

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. –

2989:

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

2183:

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

1970:

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

467:

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,

2016:

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

1749:

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

1678:

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

540:

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

463:

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

2214:

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

2073:

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

1885:

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

343:

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

162:

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

2316:

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

1693:

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

1662:

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

381:

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 -

2039:

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.

2370:

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

2339:

I'm sorry for the technical problems(Identation, signing the posts). I'm new on Knowledge. 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

488:

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

358:

I don't see how this uses the result of the Basel problem, ζ(2) = π / 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

1449:

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

1322:{\displaystyle {\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}}} 986:{\displaystyle {\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}}} 2299:

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'

1663:

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!

2017:

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.

296:

I will not go into details, but the proof starts with the function absolute value of x on the interval 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.

307:

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.

2215:

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

2147:

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 Knowledge (where the title of the corresponding article is "Problema di Basilea").

316:

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 Knowledge’s protocols.

2415:

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

2340:

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 Knowledge article that he was the first who published it. Here are the sources I mentioned :

608:. 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, 1472:

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)...

464:

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...)

227:

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.

2970:

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

2512: 2653: 1971:

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.

2946: 2738: 2975:) 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. -- 2851: 696: 749:

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.

460:

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.

2020:

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? —

324: 3019: 517: 490: 2770: 1094: 764: 1420: 140: 1081: 441:, 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. — 1957:

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

405:

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

2274:

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. –

2356:

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)

163:

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.

396:

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. --

1362: 1026: 606: 393:

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.

2566:

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:

1828:

It is a clear and unambiguous violation of our manual of style, and basically all mathematics typesetting conventions in existence. Trigonometric functions are

489:

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

248:

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?

2263: 494: 251:

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 tanx, which is : -->

2525:

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. –

261:

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).

3065: 130: 377:

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.

237:

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.

207:-series in calculus, or something. There may be some other simple ways to approximate this that people used before Euler that might be interesting. 1469:

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).

1716:

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.

1522:) -- 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/z 2548:. It is often better for our purposes to summarize a proof concisely in prose, rather than to step through it equation-by-equation. Accordingly, 2424:

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:

106: 382:

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

3060: 2561: 2230: 2434: 2303:

A single integration by substitution u = sin(t) is needed to connect the two proofs. The second exact Euler proof is currently available on

2115:

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? —

3038: 2579: 1717: 1483: 1451: 2870: 2662: 545:, 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. 2999: 2534: 2330: 2283: 2197: 2138: 1640: 1435: 328: 3023: 1332:

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

1672: 1680: 731: 97: 58: 2781: 611: 390:

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.

1659:"Because π is a transcendental number, the record holder's rectangles area will always be greater than the ideal rectangle." 252:

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.

203:

That's a good idea...I forget simple things like, "why does this sum converge in the first place", assuming everyone has seen

270: 1605: 1425:

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.

2861: 33: 2949: 2743: 2266:. They do in fact give a source, and it's similar to, but different from what you've added. In fact, I found some 722:

What is meant with "worked out in that article"? If it's the last link than the link is dead. Furthermore does the

1367: 2980: 2576:

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

2428:

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

2388: 2361: 2353: 2239: 2226: 1031: 2853:. He finally decomposes each positive integer into the product of an odd number and a power of two, then uses a 468:

right now, I think that most of our readers must stop reading somewhere in the "What you need to know" section.

2860:

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

2153: 2120: 1699: 1668: 1547: 1487: 446: 3042: 1721: 1455: 520:. This article could be more about the history of the Basel problem and maybe one proof as short as possible. 513: 1439: 2995: 2530: 2326: 2279: 2193: 2134: 2105: 1644: 560:

My understanding of zeta(2n) is that you find an nth antiderivative of f(x)=x. So for instance, assume that

2972: 2380: 1684: 735: 1619: 245:"First note that 0 < sin x < x < tan x. This can be seen by considering the following picture:" 2557: 2545: 2101: 2083: 2029: 1944: 1895: 1811: 1759: 406: 39: 83: 2984: 727: 723: 2976: 2773: 2410: 2384: 2357: 2345: 2307: 2235: 2222: 2218: 2174: 1655:

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

1636: 1589: 1479: 1431: 320: 21: 2149: 2116: 1769:

The edit that I reverted replaced \sin with {{sin}}. Yes, I object to this edit in its entirety.

1695: 1664: 442: 301: 1335: 999: 546: 498: 397: 105:

on Knowledge. If you would like to participate, please visit the project page, where you can join

2991: 2953: 2549: 2526: 2322: 2294: 2275: 2209: 2189: 2130: 1654: 1608: 1581: 1566: 563: 89: 73: 52: 2054: 1988: 1958: 1854: 1779: 1736: 1615: 1083:

since both sides equals 0 (which has been mentioned earlier in the proof). Generally we have:

550: 502: 290: 266: 3037:

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

2854: 2553: 2250: 2077: 2023: 1938: 1929: 1889: 1805: 1753: 708: 530: 478: 186: 3033:

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

2372: 1585: 2376: 2341: 174:, (since the system wouldn't let me do this directly) so this article should be deleted. 2313: 2254: 220: 2267: 3054: 2656: 2518: 2318: 2041: 1962: 1841: 1833: 1732: 1577: 1562: 542: 424: 280: 255: 229: 219:

It isn't clear to me from the article why this problem is called the Basel problem.

208: 195: 175: 171: 164: 2973:

https://en.wikipedia.org/Wikipedia_talk:These_are_not_original_research#calculations

2349: 2188:(and the rest of the page, too). Constructions like "Let's ..." are problematic. – 2967: 2417: 2180: 2068: 2047: 2011: 1981: 1920: 1880: 1847: 1797: 1772: 1744: 1576:

There's a proof using complex analysis already in the article, as the first proof:

262: 1935:

when adding to this thread, so that I receive notification of your new addition. —

512:

Nono, what I'm proposing is to segregate most of the proofs to an article, maybe,

2346:

https://fr.wikipedia.org/Probl%C3%A8me_de_B%C3%A2le#La_d%C3%A9monstration_d'Euler

726:

on wikipedia not give the desired result for this prove. Mostly because here the

704: 526: 474: 183: 102: 2317:

things before continuing to try to re-add material over objections; again, see

1491: 194:

Possibly...I find it hard to imagine someone linking to it, but it can't hurt.

2517:

by two methods: first directly, and then by expanding the inverse sine as its

2354:

https://pdfs.semanticscholar.org/5a8a/5a18e10917d364b61282eb76fd57024bdc0d.pdf

79: 360: 2544:

This is a good time to remember that we're here to write an encyclopedia,

2655:

by two methods: first directly, and then by expanding the arcsine as its

2185: 1604:

would it be too dare if someone (including me) could add the "folklore"

2381:

https://lemoid.wordpress.com/2014/03/12/basel-problem-arcsin-x-solution/

1633:

Euler's solution of the Basel problem – the longer story PDF (61.7 KB)

419:

Just a comment on the "What you need to know" sub-section... love this,

2507:{\displaystyle \int _{0}^{1}{\frac {\sin ^{-1}x}{\sqrt {1-x^{2}}}}\,dx} 438: 3046: 3027: 3003: 2648:{\displaystyle \int _{0}^{1}{\frac {\arcsin(x)}{\sqrt {1-x^{2}}}}\ dx} 2538: 2392: 2365: 2334: 2287: 2243: 2201: 2157: 2142: 2124: 2109: 2087: 2061: 2033: 1995: 1948: 1899: 1861: 1815: 1786: 1763: 1725: 1703: 1688: 1648: 1623: 1593: 1570: 1550: 1459: 1443: 739: 712: 554: 534: 506: 482: 450: 427: 409: 400: 363: 332: 283: 273: 2521:

and integrating term-by-term. Equating the two completes the proof.

352:

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

2941:{\displaystyle \int _{0}^{1}{\frac {x^{2n+1}}{\sqrt {1-x^{2}}}}\ dx} 2733:{\displaystyle \int _{0}^{1}{\frac {x^{2n+1}}{\sqrt {1-x^{2}}}}\ dx} 1844:). You need to say why ignoring the manual of style is justified. 701:

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

2304: 2258: 1561:

Isn't there a proof which uses complex analysis for the same? --

2420:. We also don't need to give the full details here. What we 2179:

The primary problem with adding this proof is that there's no

1546:). This form has its advantages, one being that b = p(0). -- 15: 2846:{\displaystyle \sum _{n=0}^{+\infty }{\frac {1}{(2n+1)^{2}}}} 691:{\displaystyle {\hat {g}}(k)={1 \over (ik)^{n}}{\hat {f}}(k)} 344:

and to suggest the best way to include this? Marc M. 20-6-06

518:

Proof that the sum of the reciprocals of the primes diverges

491:

Proof that the sum of the reciprocals of the primes diverges

1802:

And would you care to share with anyone why may that be? —

2270:

which credit this proof to a note by Boo Rim Choe in the

1450:

Weierstrass's factorization theorem in complex analysis.

2373:

https://faculty.math.illinois.edu/~reznick/sandifer.pdf

1954: 2377:

https://www.apmep.fr/IMG/pdf/Article_probleme_Bale.pdf

2342:

https://www.apmep.fr/IMG/pdf/Article_probleme_Bale.pdf

2873: 2784: 2747: 2746: 2665: 2582: 2437: 2129:

Basilea is the Italian name for Basel, apparently. –

1370: 1338: 1092: 1034: 1002: 762: 614: 566: 1840:

to justify the edit. I've already given my answer (

101:, a collaborative effort to improve the coverage of 2312:One more time, please indent and sign your posts. 1578:

Basel_problem#A_rigorous_proof_using_Fourier_series

2940: 2845: 2764: 2732: 2647: 2506: 2100:Please add a link to this from "Basilea Problem". 1414: 1356: 1321: 1075: 1020: 985: 690: 600: 2765:{\displaystyle \displaystyle n\ \in \mathbb {N} } 2350:http://eulerarchive.maa.org/hedi/HEDI-2004-03.pdf 312:Why is the same Fourier series proof given twice? 1832:typeset in roman case. But really the point of 2659:and integrating term-by-term. He then computes 300:The Riemann zeta formula can be approached by 2571: 2426: 1750:the use of itallics for the trig functions? — 1415:{\displaystyle n=\pm 1,\pm 2,\pm 3,\dots \,.} 423:helpful, should be present in more articles! 8: 2040:Please use either math tags or the template 1076:{\displaystyle n=\pm 1,\pm 2,\pm 3,\dots \,} 495:Inequality of arithmetic and geometric means 437:Would it be worthwhile to add a wikilink to 2216: 2184:math articles, please also see especially 698:and that gives you the necessary formula. 369:Fundamental Theorem of Algebra unnecessary 318: 47: 2921: 2895: 2889: 2883: 2878: 2872: 2834: 2809: 2800: 2789: 2783: 2757: 2756: 2745: 2713: 2687: 2681: 2675: 2670: 2664: 2628: 2598: 2592: 2587: 2581: 2487: 2460: 2453: 2447: 2442: 2436: 1498:For a polynomial of degree n with roots z 1369: 1337: 1292: 1261: 1230: 1199: 1173: 1147: 1125: 1097: 1093: 1091: 1033: 1001: 956: 925: 894: 863: 837: 811: 795: 767: 763: 761: 668: 667: 658: 639: 616: 615: 613: 571: 565: 279:Thanks. I've added this to the article. 2496: 2352:(The Sandifer article you mentioned) , 1407: 1071: 325:2600:1010:B002:D03B:F5FB:3BB3:90A5:C78C 182:Is it not worth keeping as a redirect? 49: 19: 3020:2001:56A:7C52:D300:884B:E679:8F43:9722 2257:your posts with 4 tildes (~~~~). See 2776:. Equating the two gives the value of 2264:WP:Knowledge is not a reliable source 1679:digit number would yet to be found). 7: 95:This article is within the scope of 718:A slicker proof from Fourier series 541:to have some proofs in the article 38:It is of interest to the following 2804: 2552:'s suggestion here is reasonable. 14: 3066:Mid-priority mathematics articles 1959:WP:MOSMATH#Multi-letter functions 289:You can prove it easily by using 115:Knowledge:WikiProject Mathematics 1476:x-pi is not the same as 1-x/pi 170:I manually renamed this article 118:Template:WikiProject Mathematics 82: 72: 51: 20: 996:Actually this is only true for 135:This article has been rated as 2831: 2815: 2613: 2607: 2233:) 16:31, 6 December 2019 (UTC) 1712:Removed square packing section 1704:03:19, 11 September 2014 (UTC) 1689:02:59, 11 September 2014 (UTC) 1624:02:33, 26 September 2012 (UTC) 1584:, also linked in the article. 1557:Proof through Complex Analysis 1112: 1106: 782: 776: 685: 679: 673: 655: 645: 633: 627: 621: 589: 583: 578: 572: 364:20:03, 27 September 2006 (UTC) 1: 3028:01:58, 30 December 2021 (UTC) 3004:18:46, 22 February 2020 (UTC) 2985:19:31, 22 December 2019 (UTC) 2864:can justify this interchange. 2562:18:23, 22 December 2019 (UTC) 2539:15:40, 22 December 2019 (UTC) 2393:10:25, 22 December 2019 (UTC) 2305:https://lemoid.wordpress.com/ 2272:American Mathematical Monthly 2158:19:55, 25 February 2018 (UTC) 2143:19:11, 25 February 2018 (UTC) 2125:19:10, 25 February 2018 (UTC) 2110:18:53, 25 February 2018 (UTC) 2088:15:08, 15 December 2015 (UTC) 2062:14:42, 15 December 2015 (UTC) 2034:14:13, 15 December 2015 (UTC) 1996:13:49, 15 December 2015 (UTC) 1949:13:46, 15 December 2015 (UTC) 1900:13:39, 15 December 2015 (UTC) 1862:13:31, 15 December 2015 (UTC) 1816:13:23, 15 December 2015 (UTC) 1787:13:19, 15 December 2015 (UTC) 1764:13:08, 15 December 2015 (UTC) 1737:WP:MOSMATH#Multi-letter names 1673:18:26, 7 September 2014 (UTC) 1580:. For many other proofs, see 1492:15:01, 19 February 2010 (UTC) 1444:14:34, 15 November 2008 (UTC) 1357:{\displaystyle x=n\cdot \pi } 1021:{\displaystyle x=n\cdot \pi } 451:22:11, 15 December 2007 (UTC) 333:07:10, 18 February 2020 (UTC) 284:23:08, 2 September 2006 (UTC) 109:and see a list of open tasks. 3061:B-Class mathematics articles 3047:18:49, 18 January 2024 (UTC) 2862:Monotone_convergence_theorem 2366:10:11, 7 December 2019 (UTC) 2335:22:27, 6 December 2019 (UTC) 2288:16:49, 6 December 2019 (UTC) 2244:16:58, 6 December 2019 (UTC) 2202:15:00, 6 December 2019 (UTC) 1726:19:22, 14 October 2015 (UTC) 1460:06:36, 16 October 2009 (UTC) 740:08:59, 8 November 2008 (UTC) 601:{\displaystyle g^{(n)}(x)=x} 274:18:18, 24 January 2006 (UTC) 2950:integration by substitution 1594:09:01, 12 August 2010 (UTC) 1571:08:53, 12 August 2010 (UTC) 373:I object to the following: 3082: 2966:The two last lines may be 1582:Robin Chapman's collection 428:03:20, 6 August 2007 (UTC) 410:17:41, 7 August 2007 (UTC) 223:19:49, Aug 22, 2004 (UTC) 1649:20:51, 12 June 2013 (UTC) 1607:? it is about some other 713:23:44, 24 July 2008 (UTC) 555:08:06, 24 July 2008 (UTC) 535:02:18, 24 July 2008 (UTC) 507:00:37, 24 July 2008 (UTC) 483:19:55, 23 July 2008 (UTC) 401:08:35, 27 July 2007 (UTC) 189:02:18, Feb 29, 2004 (UTC) 178:20:28, 26 Feb 2004 (UTC) 167:15:31, 26 Feb 2004 (UTC) 134: 67: 46: 1551:16:45, 4 July 2010 (UTC) 514:Proof that zeta(2)=π^2/6 232:17:51, 24 Aug 2004 (UTC) 215:Overlooking the obvious? 211:22:18, 3 Mar 2004 (UTC) 198:02:41, 29 Feb 2004 (UTC) 141:project's priority scale 1836:is that the onus is on 98:WikiProject Mathematics 2961: 2942: 2857:to complete the proof. 2847: 2808: 2766: 2734: 2649: 2523: 2508: 1465:Flaw in Euler's method 1464: 1416: 1358: 1323: 1077: 1022: 987: 692: 602: 386:) over all the roots r 379: 339:Basel problem extended 28:This article is rated 2943: 2867:One can also compute 2848: 2785: 2767: 2735: 2650: 2509: 2261:for more information. 1925:Also, please use the 1417: 1359: 1324: 1078: 1023: 988: 693: 603: 415:What you need to know 375: 2871: 2782: 2774:integration by parts 2744: 2663: 2580: 2435: 1611:' solved problem :) 1368: 1336: 1090: 1032: 1000: 760: 612: 564: 121:mathematics articles 2888: 2680: 2597: 2452: 1953:Ok, which parts of 753:It is stated that: 728:Parseval's identity 724:Parseval's identity 2938: 2874: 2843: 2762: 2761: 2730: 2666: 2645: 2583: 2504: 2497: 2438: 1609:sum of reciprocals 1412: 1408: 1354: 1319: 1317: 1073: 1072: 1018: 983: 981: 688: 598: 302:Parseval's theorem 90:Mathematics portal 34:content assessment 2954:Wallis' integrals 2931: 2928: 2927: 2841: 2752: 2723: 2720: 2719: 2638: 2635: 2634: 2494: 2493: 2268:notes from a talk 2262: 2234: 2221:comment added by 2060: 1994: 1860: 1785: 1639:comment added by 1534:) where b = a (-z 1482:comment added by 1434:comment added by 1305: 1274: 1243: 1212: 1181: 1155: 1119: 969: 938: 907: 876: 845: 819: 789: 745:Problem in proof. 730:is stated wrong. 676: 665: 624: 335: 323:comment added by 291:Fourier transform 263:Fredrik Johansson 155: 154: 151: 150: 147: 146: 3073: 2947: 2945: 2944: 2939: 2930: 2929: 2926: 2925: 2910: 2909: 2908: 2890: 2887: 2882: 2855:geometric series 2852: 2850: 2849: 2844: 2842: 2840: 2839: 2838: 2810: 2807: 2799: 2771: 2769: 2768: 2763: 2760: 2751: 2739: 2737: 2736: 2731: 2722: 2721: 2718: 2717: 2702: 2701: 2700: 2682: 2679: 2674: 2654: 2652: 2651: 2646: 2637: 2636: 2633: 2632: 2617: 2616: 2599: 2596: 2591: 2513: 2511: 2510: 2505: 2495: 2492: 2491: 2476: 2475: 2468: 2467: 2454: 2451: 2446: 2414: 2298: 2248: 2213: 2178: 2072: 2059: 2057: 2045: 2015: 1993: 1991: 1979: 1934: 1928: 1924: 1884: 1859: 1857: 1845: 1801: 1784: 1782: 1770: 1748: 1651: 1494: 1446: 1421: 1419: 1418: 1413: 1363: 1361: 1360: 1355: 1328: 1326: 1325: 1320: 1318: 1311: 1307: 1306: 1304: 1293: 1280: 1276: 1275: 1273: 1262: 1249: 1245: 1244: 1242: 1231: 1218: 1214: 1213: 1211: 1200: 1187: 1183: 1182: 1174: 1161: 1157: 1156: 1148: 1126: 1120: 1115: 1098: 1082: 1080: 1079: 1074: 1027: 1025: 1024: 1019: 992: 990: 989: 984: 982: 975: 971: 970: 968: 957: 944: 940: 939: 937: 926: 913: 909: 908: 906: 895: 882: 878: 877: 875: 864: 851: 847: 846: 838: 825: 821: 820: 812: 796: 790: 785: 768: 697: 695: 694: 689: 678: 677: 669: 666: 664: 663: 662: 640: 626: 625: 617: 607: 605: 604: 599: 582: 581: 471:Any objections? 348:Irrelevant topic 241:Picture as proof 123: 122: 119: 116: 113: 92: 87: 86: 76: 69: 68: 63: 55: 48: 31: 25: 24: 16: 3081: 3080: 3076: 3075: 3074: 3072: 3071: 3070: 3051: 3050: 3035: 3016: 2977:Contribute.Math 2917: 2891: 2869: 2868: 2830: 2814: 2780: 2779: 2742: 2741: 2709: 2683: 2661: 2660: 2624: 2600: 2578: 2577: 2483: 2456: 2455: 2433: 2432: 2411:Contribute.Math 2408: 2385:Contribute.Math 2358:Contribute.Math 2308:Contribute.Math 2292: 2236:Contribute.Math 2223:Contribute.Math 2207: 2175:Contribute.Math 2172: 2170: 2168:Recent addition 2098: 2096:Basilea Problem 2066: 2055: 2050: 2046: 2009: 1989: 1984: 1980: 1932: 1926: 1918: 1878: 1855: 1850: 1846: 1795: 1780: 1775: 1771: 1742: 1740: 1731:Formatting per 1714: 1657: 1634: 1631: 1602: 1600:wouldt it be... 1559: 1545: 1541: 1537: 1533: 1529: 1525: 1521: 1517: 1513: 1510:; p(z) = a (z-z 1509: 1505: 1501: 1477: 1467: 1429: 1366: 1365: 1334: 1333: 1316: 1315: 1297: 1285: 1281: 1266: 1254: 1250: 1235: 1223: 1219: 1204: 1192: 1188: 1166: 1162: 1140: 1136: 1121: 1099: 1088: 1087: 1030: 1029: 998: 997: 980: 979: 961: 949: 945: 930: 918: 914: 899: 887: 883: 868: 856: 852: 830: 826: 804: 800: 791: 769: 758: 757: 747: 720: 654: 644: 610: 609: 567: 562: 561: 458: 435: 417: 389: 385: 371: 350: 341: 314: 294: 243: 217: 160: 120: 117: 114: 111: 110: 88: 81: 61: 32:on Knowledge's 29: 12: 11: 5: 3079: 3077: 3069: 3068: 3063: 3053: 3052: 3039:146.95.224.227 3034: 3031: 3018:Noice article 3015: 3012: 3011: 3010: 3009: 3008: 3007: 3006: 2960: 2959: 2958: 2957: 2952:and recognize 2937: 2934: 2924: 2920: 2916: 2913: 2907: 2904: 2901: 2898: 2894: 2886: 2881: 2877: 2865: 2858: 2837: 2833: 2829: 2826: 2823: 2820: 2817: 2813: 2806: 2803: 2798: 2795: 2792: 2788: 2777: 2759: 2755: 2750: 2729: 2726: 2716: 2712: 2708: 2705: 2699: 2696: 2693: 2690: 2686: 2678: 2673: 2669: 2644: 2641: 2631: 2627: 2623: 2620: 2615: 2612: 2609: 2606: 2603: 2595: 2590: 2586: 2570: 2569: 2568: 2567: 2546:not a textbook 2515: 2514: 2503: 2500: 2490: 2486: 2482: 2479: 2474: 2471: 2466: 2463: 2459: 2450: 2445: 2441: 2423: 2406: 2405: 2404: 2403: 2402: 2401: 2400: 2399: 2398: 2397: 2396: 2395: 2301: 2169: 2166: 2165: 2164: 2163: 2162: 2161: 2160: 2150:David Eppstein 2117:David Eppstein 2097: 2094: 2093: 2092: 2091: 2090: 2053: 2048: 2007: 2006: 2005: 2004: 2003: 2002: 2001: 2000: 1999: 1998: 1987: 1982: 1976: 1975: 1974: 1973: 1972: 1909: 1908: 1907: 1906: 1905: 1904: 1903: 1902: 1869: 1868: 1867: 1866: 1865: 1864: 1853: 1848: 1821: 1820: 1819: 1818: 1790: 1789: 1778: 1773: 1739: 1729: 1718:24.144.122.240 1713: 1710: 1709: 1708: 1707: 1706: 1696:David Eppstein 1656: 1653: 1630: 1627: 1601: 1598: 1597: 1596: 1558: 1555: 1554: 1553: 1543: 1539: 1535: 1531: 1527: 1523: 1519: 1515: 1511: 1507: 1503: 1499: 1484:76.182.194.195 1466: 1463: 1452:97.127.142.123 1428: 1411: 1406: 1403: 1400: 1397: 1394: 1391: 1388: 1385: 1382: 1379: 1376: 1373: 1353: 1350: 1347: 1344: 1341: 1330: 1329: 1314: 1310: 1303: 1300: 1296: 1291: 1288: 1284: 1279: 1272: 1269: 1265: 1260: 1257: 1253: 1248: 1241: 1238: 1234: 1229: 1226: 1222: 1217: 1210: 1207: 1203: 1198: 1195: 1191: 1186: 1180: 1177: 1172: 1169: 1165: 1160: 1154: 1151: 1146: 1143: 1139: 1135: 1132: 1129: 1124: 1122: 1118: 1114: 1111: 1108: 1105: 1102: 1096: 1095: 1070: 1067: 1064: 1061: 1058: 1055: 1052: 1049: 1046: 1043: 1040: 1037: 1017: 1014: 1011: 1008: 1005: 994: 993: 978: 974: 967: 964: 960: 955: 952: 948: 943: 936: 933: 929: 924: 921: 917: 912: 905: 902: 898: 893: 890: 886: 881: 874: 871: 867: 862: 859: 855: 850: 844: 841: 836: 833: 829: 824: 818: 815: 810: 807: 803: 799: 794: 792: 788: 784: 781: 778: 775: 772: 766: 765: 746: 743: 719: 716: 687: 684: 681: 675: 672: 661: 657: 653: 650: 647: 643: 638: 635: 632: 629: 623: 620: 597: 594: 591: 588: 585: 580: 577: 574: 570: 558: 557: 510: 509: 457: 454: 443:David Eppstein 434: 431: 416: 413: 387: 383: 370: 367: 349: 346: 340: 337: 313: 310: 293: 287: 277: 276: 242: 239: 236: 234: 233: 216: 213: 201: 200: 199: 191: 190: 159: 156: 153: 152: 149: 148: 145: 144: 133: 127: 126: 124: 107:the discussion 94: 93: 77: 65: 64: 56: 44: 43: 37: 26: 13: 10: 9: 6: 4: 3: 2: 3078: 3067: 3064: 3062: 3059: 3058: 3056: 3049: 3048: 3044: 3040: 3032: 3030: 3029: 3025: 3021: 3013: 3005: 3001: 2997: 2993: 2992:Deacon Vorbis 2988: 2987: 2986: 2982: 2978: 2974: 2969: 2965: 2964: 2963: 2962: 2955: 2951: 2935: 2932: 2922: 2918: 2914: 2911: 2905: 2902: 2899: 2896: 2892: 2884: 2879: 2875: 2866: 2863: 2859: 2856: 2835: 2827: 2824: 2821: 2818: 2811: 2801: 2796: 2793: 2790: 2786: 2778: 2775: 2753: 2748: 2727: 2724: 2714: 2710: 2706: 2703: 2697: 2694: 2691: 2688: 2684: 2676: 2671: 2667: 2658: 2657:Taylor series 2642: 2639: 2629: 2625: 2621: 2618: 2610: 2604: 2601: 2593: 2588: 2584: 2575: 2574: 2573: 2572: 2565: 2564: 2563: 2559: 2555: 2551: 2550:Deacon Vorbis 2547: 2543: 2542: 2541: 2540: 2536: 2532: 2528: 2527:Deacon Vorbis 2522: 2520: 2519:Taylor series 2501: 2498: 2488: 2484: 2480: 2477: 2472: 2469: 2464: 2461: 2457: 2448: 2443: 2439: 2431: 2430: 2429: 2425: 2421: 2419: 2412: 2394: 2390: 2386: 2382: 2378: 2374: 2369: 2368: 2367: 2363: 2359: 2355: 2351: 2347: 2343: 2338: 2337: 2336: 2332: 2328: 2324: 2323:Deacon Vorbis 2320: 2315: 2311: 2310: 2309: 2306: 2302: 2296: 2295:Deacon Vorbis 2291: 2290: 2289: 2285: 2281: 2277: 2276:Deacon Vorbis 2273: 2269: 2265: 2260: 2256: 2252: 2247: 2246: 2245: 2241: 2237: 2232: 2228: 2224: 2220: 2211: 2210:Deacon Vorbis 2206: 2205: 2204: 2203: 2199: 2195: 2191: 2190:Deacon Vorbis 2187: 2186:MOS:MATH#TONE 2182: 2176: 2167: 2159: 2155: 2151: 2146: 2145: 2144: 2140: 2136: 2132: 2131:Deacon Vorbis 2128: 2127: 2126: 2122: 2118: 2114: 2113: 2112: 2111: 2107: 2103: 2102:John G Hasler 2095: 2089: 2085: 2081: 2080: 2079: 2070: 2065: 2064: 2063: 2058: 2052: 2043: 2042:Template:Math 2038: 2037: 2036: 2035: 2031: 2027: 2026: 2025: 2018: 2013: 1997: 1992: 1986: 1977: 1969: 1968: 1967: 1966: 1964: 1960: 1956: 1952: 1951: 1950: 1946: 1942: 1941: 1940: 1931: 1922: 1917: 1916: 1915: 1914: 1913: 1912: 1911: 1910: 1901: 1897: 1893: 1892: 1891: 1882: 1877: 1876: 1875: 1874: 1873: 1872: 1871: 1870: 1863: 1858: 1852: 1843: 1839: 1835: 1831: 1827: 1826: 1825: 1824: 1823: 1822: 1817: 1813: 1809: 1808: 1807: 1799: 1794: 1793: 1792: 1791: 1788: 1783: 1777: 1768: 1767: 1766: 1765: 1761: 1757: 1756: 1755: 1746: 1738: 1734: 1730: 1728: 1727: 1723: 1719: 1711: 1705: 1701: 1697: 1692: 1691: 1690: 1686: 1682: 1677: 1676: 1675: 1674: 1670: 1666: 1665:Robert Walker 1660: 1652: 1650: 1646: 1642: 1641:97.86.236.127 1638: 1628: 1626: 1625: 1621: 1617: 1612: 1610: 1606: 1599: 1595: 1591: 1587: 1583: 1579: 1575: 1574: 1573: 1572: 1568: 1564: 1556: 1552: 1549: 1497: 1496: 1495: 1493: 1489: 1485: 1481: 1474: 1470: 1462: 1461: 1457: 1453: 1447: 1445: 1441: 1437: 1436:82.211.210.23 1433: 1426: 1423: 1409: 1404: 1401: 1398: 1395: 1392: 1389: 1386: 1383: 1380: 1377: 1374: 1371: 1351: 1348: 1345: 1342: 1339: 1312: 1308: 1301: 1298: 1294: 1289: 1286: 1282: 1277: 1270: 1267: 1263: 1258: 1255: 1251: 1246: 1239: 1236: 1232: 1227: 1224: 1220: 1215: 1208: 1205: 1201: 1196: 1193: 1189: 1184: 1178: 1175: 1170: 1167: 1163: 1158: 1152: 1149: 1144: 1141: 1137: 1133: 1130: 1127: 1123: 1116: 1109: 1103: 1100: 1086: 1085: 1084: 1068: 1065: 1062: 1059: 1056: 1053: 1050: 1047: 1044: 1041: 1038: 1035: 1015: 1012: 1009: 1006: 1003: 976: 972: 965: 962: 958: 953: 950: 946: 941: 934: 931: 927: 922: 919: 915: 910: 903: 900: 896: 891: 888: 884: 879: 872: 869: 865: 860: 857: 853: 848: 842: 839: 834: 831: 827: 822: 816: 813: 808: 805: 801: 797: 793: 786: 779: 773: 770: 756: 755: 754: 751: 744: 742: 741: 737: 733: 729: 725: 717: 715: 714: 710: 706: 702: 699: 682: 670: 659: 651: 648: 641: 636: 630: 618: 595: 592: 586: 575: 568: 556: 552: 548: 544: 543:zeta constant 539: 538: 537: 536: 532: 528: 524: 521: 519: 515: 508: 504: 500: 496: 492: 487: 486: 485: 484: 480: 476: 472: 469: 465: 461: 455: 453: 452: 448: 444: 440: 432: 430: 429: 426: 422: 414: 412: 411: 408: 407:70.118.127.94 403: 402: 399: 394: 391: 378: 374: 368: 366: 365: 362: 356: 353: 347: 345: 338: 336: 334: 330: 326: 322: 311: 309: 305: 303: 298: 292: 288: 286: 285: 282: 275: 272: 268: 264: 260: 259: 258: 257: 253: 249: 246: 240: 238: 231: 226: 225: 224: 222: 214: 212: 210: 206: 197: 193: 192: 188: 185: 181: 180: 179: 177: 173: 172:Basel problem 168: 166: 158:Headline text 157: 142: 138: 132: 129: 128: 125: 108: 104: 100: 99: 91: 85: 80: 78: 75: 71: 70: 66: 60: 57: 54: 50: 45: 41: 35: 27: 23: 18: 17: 3036: 3017: 2524: 2516: 2427: 2407: 2344:(pp 17-19), 2271: 2217:— Preceding 2171: 2099: 2076: 2075: 2022: 2021: 2019: 2008: 1937: 1936: 1888: 1887: 1837: 1829: 1804: 1803: 1752: 1751: 1741: 1715: 1661: 1658: 1635:— Preceding 1632: 1613: 1603: 1560: 1530:) ... (1-z/z 1475: 1471: 1468: 1448: 1427: 1424: 1331: 995: 752: 748: 721: 703: 700: 559: 525: 522: 516:, much like 511: 473: 470: 466: 462: 459: 436: 420: 418: 404: 395: 392: 380: 376: 372: 357: 354: 351: 342: 319:— Preceding 315: 306: 299: 295: 278: 254: 250: 247: 244: 235: 218: 204: 202: 169: 161: 137:Mid-priority 136: 96: 62:Mid‑priority 40:WikiProjects 2078:Boruch Baum 2024:Boruch Baum 1939:Boruch Baum 1890:Boruch Baum 1806:Boruch Baum 1754:Boruch Baum 1681:98.95.5.178 1629:broken link 1478:—Preceding 1430:—Preceding 732:86.95.192.7 112:Mathematics 103:mathematics 59:Mathematics 3055:Categories 2554:XOR'easter 2300:integrals. 1586:Shreevatsa 1518:) ... (z-z 456:Long proof 2948:using an 2772:using an 2251:WP:INDENT 1955:this edit 1542:) ... (-z 421:extremely 221:Dataphile 3014:Feedback 2314:WP:PINGs 2231:contribs 2219:unsigned 1637:unsigned 1563:Hirak 99 1526:) (1-z/z 1506:, ..., z 1480:unsigned 1432:unsigned 433:Coprime? 425:Error792 321:unsigned 281:Buster79 271:contribs 256:Buster79 230:Revolver 209:Revolver 196:Revolver 176:Revolver 165:Revolver 2255:WP:SIGN 2249:Please 2074:made. — 2069:Slawekb 2051:ławomir 2012:Slawekb 1985:ławomir 1930:replyto 1921:Slawekb 1881:Slawekb 1851:ławomir 1798:Slawekb 1776:ławomir 1745:Slawekb 547:Schmock 499:Schmock 439:coprime 398:Gene496 139:on the 30:B-class 3000:videos 2996:carbon 2602:arcsin 2535:videos 2531:carbon 2331:videos 2327:carbon 2319:WP:BRD 2284:videos 2280:carbon 2198:videos 2194:carbon 2181:source 2139:videos 2135:carbon 1963:WP:MSM 1842:WP:MSM 1834:WP:BRD 1830:always 1733:WP:MSM 1514:) (z-z 1364:where 1028:where 705:Loisel 527:Loisel 475:Loisel 184:Angela 36:scale. 2968:WP:OR 2740:with 2422:could 2418:WP:OR 2056:Biały 1990:Biały 1856:Biały 1781:Biały 1616:kmath 1538:) (-z 3043:talk 3024:talk 2981:talk 2558:talk 2389:talk 2362:talk 2321:. – 2259:H:TP 2253:and 2240:talk 2227:talk 2154:talk 2121:talk 2106:talk 2084:talk 2030:talk 1945:talk 1896:talk 1812:talk 1760:talk 1735:and 1722:talk 1700:talk 1685:talk 1669:talk 1645:talk 1620:talk 1590:talk 1567:talk 1488:talk 1456:talk 1440:talk 736:talk 709:talk 551:talk 531:talk 523:No? 503:talk 493:and 479:talk 447:talk 361:Gwil 329:talk 267:talk 2458:sin 1838:you 1548:ToE 1502:, z 1101:sin 771:sin 497:. 131:Mid 3057:: 3045:) 3026:) 3002:) 2998:• 2983:) 2915:− 2876:∫ 2805:∞ 2787:∑ 2754:∈ 2707:− 2668:∫ 2622:− 2605:⁡ 2585:∫ 2560:) 2537:) 2533:• 2481:− 2470:⁡ 2462:− 2440:∫ 2391:) 2379:, 2375:, 2364:) 2348:, 2333:) 2329:• 2286:) 2282:• 2242:) 2229:• 2200:) 2196:• 2156:) 2141:) 2137:• 2123:) 2108:) 2086:) 2044:. 2032:) 1978:-- 1965:: 1947:) 1933:}} 1927:{{ 1898:) 1814:) 1762:) 1724:) 1702:) 1687:) 1671:) 1647:) 1622:) 1614:-- 1592:) 1569:) 1490:) 1458:) 1442:) 1422:. 1405:… 1396:± 1387:± 1378:± 1352:π 1349:⋅ 1313:⋯ 1302:π 1271:π 1259:− 1240:π 1209:π 1197:− 1179:π 1153:π 1145:− 1134:⋅ 1104:⁡ 1069:… 1060:± 1051:± 1042:± 1016:π 1013:⋅ 977:⋯ 966:π 935:π 923:− 904:π 873:π 861:− 843:π 817:π 809:− 774:⁡ 738:) 711:) 674:^ 622:^ 553:) 533:) 505:) 481:) 449:) 331:) 304:. 269:- 265:- 3041:( 3022:( 2994:( 2979:( 2956:. 2936:x 2933:d 2923:2 2919:x 2912:1 2906:1 2903:+ 2900:n 2897:2 2893:x 2885:1 2880:0 2836:2 2832:) 2828:1 2825:+ 2822:n 2819:2 2816:( 2812:1 2802:+ 2797:0 2794:= 2791:n 2758:N 2749:n 2728:x 2725:d 2715:2 2711:x 2704:1 2698:1 2695:+ 2692:n 2689:2 2685:x 2677:1 2672:0 2643:x 2640:d 2630:2 2626:x 2619:1 2614:) 2611:x 2608:( 2594:1 2589:0 2556:( 2529:( 2502:x 2499:d 2489:2 2485:x 2478:1 2473:x 2465:1 2449:1 2444:0 2413:: 2409:@ 2387:( 2360:( 2325:( 2297:: 2293:@ 2278:( 2238:( 2225:( 2212:: 2208:@ 2192:( 2177:: 2173:@ 2152:( 2148:— 2133:( 2119:( 2104:( 2082:( 2071:: 2067:@ 2049:S 2028:( 2014:: 2010:@ 1983:S 1943:( 1923:: 1919:@ 1894:( 1883:: 1879:@ 1849:S 1810:( 1800:: 1796:@ 1774:S 1758:( 1747:: 1743:@ 1720:( 1698:( 1683:( 1667:( 1643:( 1618:( 1588:( 1565:( 1544:n 1540:2 1536:1 1532:n 1528:2 1524:1 1520:n 1516:2 1512:1 1508:n 1504:2 1500:1 1486:( 1454:( 1438:( 1410:. 1402:, 1399:3 1393:, 1390:2 1384:, 1381:1 1375:= 1372:n 1346:n 1343:= 1340:x 1309:) 1299:3 1295:x 1290:+ 1287:1 1283:( 1278:) 1268:3 1264:x 1256:1 1252:( 1247:) 1237:2 1233:x 1228:+ 1225:1 1221:( 1216:) 1206:2 1202:x 1194:1 1190:( 1185:) 1176:x 1171:+ 1168:1 1164:( 1159:) 1150:x 1142:1 1138:( 1131:a 1128:= 1117:x 1113:) 1110:x 1107:( 1066:, 1063:3 1057:, 1054:2 1048:, 1045:1 1039:= 1036:n 1010:n 1007:= 1004:x 973:) 963:3 959:x 954:+ 951:1 947:( 942:) 932:3 928:x 920:1 916:( 911:) 901:2 897:x 892:+ 889:1 885:( 880:) 870:2 866:x 858:1 854:( 849:) 840:x 835:+ 832:1 828:( 823:) 814:x 806:1 802:( 798:= 787:x 783:) 780:x 777:( 734:( 707:( 686:) 683:k 680:( 671:f 660:n 656:) 652:k 649:i 646:( 642:1 637:= 634:) 631:k 628:( 619:g 596:x 593:= 590:) 587:x 584:( 579:) 576:n 573:( 569:g 549:( 529:( 501:( 477:( 445:( 388:j 384:j 327:( 205:p 187:. 143:. 42::

Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.

Index