Talk:Leibniz formula for π

Leibniz formula for pi
The justification of the term-by-term integration here is not actually trivial. Charles Matthews 12:34, 9 Mar 2005 (UTC)

Alternative summations
$$1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} ...$$

can be rewritten as:

$$1 + \frac{1}{3} + \frac{1}{5} + \frac{1}{7} ... - \frac{2}{3} - \frac{2}{7} - \frac{2}{9} ...$$

ie: as the sum of two overlapping infinite summations, where one is all positive and the other is all negative.

This necessarily converges far worse than the original formulation, as the two sums step at different rates. It works only because the infinities are both of natural numbers and thus equal infinities, so it doesn't matter that the step rate is different. But continuing with it anyway, we get:

$$\sum_{n=1}^\infty \frac{1}{2n - 1} - \sum_{n=1}^\infty \frac{2}{4n - 1}$$

Reduce to a single summation:

$$= \sum_{n=1}^\infty (\frac{1}{2n - 1} - \frac{2}{4n - 1})$$

Get the denominators equal:

$$= \sum_{n=1}^\infty (\frac{4n - 1}{(2n - 1)(4n - 1)} - \frac{4n - 2}{(2n - 1)(4n - 1)})$$

Combine the two fractions:

$$= \sum_{n=1}^\infty \frac{(4n - 1) - (4n - 2)}{(2n - 1)(4n - 1)}$$

Simplify:

$$= \sum_{n=1}^\infty \frac{1}{(2n - 1)(4n - 1)}$$

Since Pi is equal to four times this, we get that Pi then equals:

$$4\sum_{n=1}^\infty \frac{1}{(2n - 1)(4n - 1)}$$

As best as I can tell, this summation is completely useless, but seems easier to scan with the eye than a lot I've seen. —Preceding unsigned comment added by 75.164.145.226 (talk) 04:07, 17 February 2008 (UTC)


 * Since the series is only conditionally and not absolutely convergent, this whole remark is rubbish from the start. Reordering a conditonally convergent series allows to converge to any real number or to diverge.--LutzL (talk) 19:00, 12 May 2011 (UTC)


 * using same start point and step

$$\sum_{n=1}^\infty \frac{1}{4n - 3} - \sum_{n=1}^\infty \frac{1}{4n - 1}$$ simplifies to $$\sum_{n=1}^\infty \frac{2}{16n^2-16n + 3}$$ 2600:1003:B84B:D477:14A6:A3D9:895A:5D12 (talk) 18:23, 12 March 2021 (UTC)

Better proof
I was trying to find &pi; using a power series in one way or another in order to get the Leibniz formula, and I found this to be a better method of showing how:

We know that $$\arctan 1 = \frac{\pi}{4}$$

Take the first derivative of arctan(x) and put it in a geometric series:

$$\frac{d}{dx} \arctan x = \frac{1}{1 + x^2} = \sum_{n=1}^\infty (-x^2)^{n-1} = \sum_{n=1}^\infty (-1)^{n-1} x^{2n-2}$$

Integrate that:

$$\int_1^x (-1)^{n-1} t^{2n-2} \, dt = \frac{(-1)^{n-1}}{2n-1} x^{2n-1}$$

So, we have a power series for arctan(x):

$$\arctan x = \sum_{n=1}^\infty \frac{(-1)^{n-1}}{2n-1} x^{2n-1}$$

Plug in 1 for x and get &pi;/4:

$$\arctan 1 = \sum_{n=1}^\infty \frac{(-1)^{n-1}}{2n-1} = \frac{\pi}{4}$$

For aesthetics (article uses n=0 to &infin;):

$$\arctan 1 = \sum_{n=0}^\infty \frac{(-1)^n}{2n+1}$$

Q.E.D.

-Matt 20:11, 18 March 2006 (UTC)

"However, if the series is truncated at the right time, the decimal expansion of the approximation will agree with that of π for many more digits, except for isolated digits or digit groups."

I feel there should be more elaboration on this.


 * Hint: if N is a power of ten, each term in the right sum will be a finite decimal fraction. I agree that there is room for elaboration in the article. Feel free to edit it. Fredrik Johansson 11:29, 13 September 2006 (UTC)

Title?
Is there a reason that this article is not at Leibniz series? Srnec (talk) 05:53, 16 November 2008 (UTC)

Whats with the *systematic* attempt to rename every mathematical formula or series as the "Madhava-xxx" formula or series? There's only one book reference to this Kerala school of mathematics. and no references from any peer reviewed journal that the Leibniz Series should be renamed. The damage from this oversight of review is becoming extensive as numerous articles on the web are referring to this one in incorrectly naming it.

Seriously, if the claims made by the author were even half valid then Indian mathematics would have allowed us to have worm-hole technology by now. Xp fun (talk)
 * Can anyone comment on the validity of these claims? I can try to suggest a less controversial rewrite of the article, also is there a better forum than this area for such broad changes? Xp fun (talk) 21:44, 14 August 2009 (UTC)


 * "You can wake up a person who is sleeping, but never a person who pretends to do so!"
 * 1. If a book, or a source is not "available in Google Books or Scholar" does it mean it does not exist?!
 * 2. If you clicked through the (presently ALSO ill-named) "Gregory Series" you will see 4 references to books published in the 1500s in India. Gregory's floruit was 1668.
 * 3. It is a fact that Madhava's floruit is 2 FURTHER centuries older than that.
 * 4. STOP the blatant WHITEWASHING AND RACISM! Stop the BLATANT EUROCENTRISM.
 * Thanks! Peace Out!! Shivasundar (talk) 23:13, 14 April 2022 (UTC)

I found this paragraph on Google books, from Mathematics in India by Kim Plofker.

However, I should point out that "Madhava–Leibniz series" has only two hits on Google books and one hit on Google scholar, all of which are references specifically about Indian mathematics. I see no evidence that the term "Madhava–Leibniz series" is used by mathematicians, and Wikipedia is not an appropriate forum for advocating a more correct name for a mathematical concept. Since the material is of historical interest, I have moved it from the lead to a new section on history. Jim (talk) 16:44, 23 October 2009 (UTC)

Gregory-Leibniz vs. Leibniz-Gregory
It seems that "Gregory-Leibniz" is the more common ordering. In particular, "Gregory-Leibniz series" gets while "Leibniz-Gregory series" only gets Jim (talk) 16:28, 23 October 2009 (UTC)
 * 27,700 Google hits,
 * 41 hits on Google books, and
 * 23 hits on Google scholar,
 * 9,560 Google hits,
 * 8 hits on Google books, and
 * 4 hits on Google scholar.

POV History?
"To give the rightful place to this great mathematician, the series should be named 'mAdhava srENi' or Madhava Series. But now Madhava's work is sometimes known as the Madhava–Leibniz series."

Great? Rightful place? Should? "But now"? I believe it would be better suited for this articule to discuss other names the series could be known by in the opening paragraph (with redirections as necessary). History should include all of the history of the series rather than just the history of one "great" mathematician over others.

-Ben0mega (talk) 00:08, 19 May 2010 (UTC)

an edit
I edited the article to make it shorter and less controversial. I removed the part of the proof that was not a proof. I removed wishful thinking about honoring Madhava. I changed the heading 'history' to 'names'. Bo Jacoby (talk) 06:22, 1 June 2010 (UTC).

The reference: George E. Andrews, Richard Askey, Ranjan Roy (1999), Special Functions, Cambridge University Press, p. 58, ISBN 0521789885 says (in an exercise on p. 59 actually) that the formula was known to Madhava, but not that it is called the Madhava-Leibnitz formula. I will correct the article. Bo Jacoby (talk) 13:25, 6 June 2010 (UTC).

Requested move part 1

 * The following discussion is an archived discussion of a requested move. Please do not modify it. Subsequent comments should be made in a new section on the talk page. No further edits should be made to this section. 

The result of the move request was: no consensus for move, partially in light of Talk:Pi  Skomorokh   14:43, 12 May 2011 (UTC)

Leibniz formula for pi → Leibniz formula for π – Consistency with other similar articles per recent discussions at WPMATH and elsewhere. Ben (talk) 08:12, 4 May 2011 (UTC)


 * Support. To prevent confusion with Leibniz' famous apple pie recipe. Hans Adler 08:51, 4 May 2011 (UTC)
 * Oppose. The consistency argument is illogical when the main title is Pi. Dictionaries are the authority on spelling and they all use "pi". What about the other Greek letters? They are almost always spelled out in titles: Alpha beta transformation, Dirac delta function, Gamma function, chi squared distribution, Omega constant, etc, etc, etc. Kauffner (talk) 15:25, 7 May 2011 (UTC)
 * You never give up, do you? All the articles that you just mentioned are named after terms, per WP:COMMON. The articles that you have disruptively moved from π to pi have descriptive names, and for such titles "pi" is plain weird and eccentric. And this is a matter of mathematical typography, not spelling. Dictionaries are totally unqualified for that. Hans Adler 16:37, 7 May 2011 (UTC)
 * Support. This issue has been widely discussed elsewhere and consensus among people dealing with maths is that the symbol is a better look for non-defining articles. Xanthoxyl  &lt; 16:33, 8 May 2011 (UTC)
 * The above discussion is preserved as an archive of a requested move. Please do not modify it. Subsequent comments should be made in a new section on this talk page. No further edits should be made to this section.

Requested move part 2

 * The following discussion is an archived discussion of a requested move. Please do not modify it. Subsequent comments should be made in a new section on the talk page. No further edits should be made to this section. 

The result of the move request was: Move.  — Edokter  ( talk ) — 12:23, 8 June 2011 (UTC)

Leibniz formula for pi → Leibniz formula for π – Reopening the requested move following discussion at Wikipedia talk:WikiProject Mathematics. The early non admin closure was not per policy as there was not an overriding consensus. Salix (talk): 06:45, 13 May 2011 (UTC)
 * Comment there is some weight to be given to Talk:Pi where the main article is Pi, however looking at Category:Pi it seems that most articles use π unless there is a particular reason not to. So we have Approximations of π and Pi Day where the latter is the actual name of the day.--Salix (talk): 06:45, 13 May 2011 (UTC)
 * Oppose I just prefere titles I can type easily.--Salix (talk): 06:45, 13 May 2011 (UTC)
 * By that argument, should we move WikiProject Mathematics to WP:WPMATH? Ben (talk) 07:46, 13 May 2011 (UTC)


 * Support. I can type π easily and while I prefer words to notation in titles I also prefer notation to spelled-out-in-letters substitutes for notation. —David Eppstein (talk) 07:27, 13 May 2011 (UTC)
 * Support. It's the way it's written in mathematics.  Leibniz formula for pi will redirect to Leibniz formula for π, so that, even if you can't type "π", you can still get to the article.  — Arthur Rubin  (talk) 07:36, 13 May 2011 (UTC)
 * Support. Consistency with other similar articles per recent discussions at WPMATH and elsewhere. Ben (talk) 07:46, 13 May 2011 (UTC)
 * Oppose. The fact that the previous RM was closed a day early is being used as a pretext for an immediate reopening. But who thinks that another day would have changed the outcome? This is just bad faith. In RMs, the word "consistency" usually suggests that usage in descriptive titles should correspond to that in the main title, which in this case is Pi. The dictionaries all give this word as "pi" -- no spelling variants, no notations about math usage. Published reference works don't use special characters in article titles, see the Britannica entry on "pi". If using the pi symbol is a narrowly math usage, that would seem to be a reason for Wiki not to use it. But in fact it is easy to come up with examples of math titles that use spell-out pi. Here are 13 since 1980. Kauffner (talk) 08:46, 13 May 2011 (UTC)
 * Please note that this user has !voted before in part 1 (which was closed prematurely). I don't see any reason to !vote again, as I expect that the previous !votes will be taken into account. The problem wasn't just the the closing before the discussion had settled completely, the problem was closing it with a rationale that was a !vote rationale, not a closing rationale, and turning a 3:1 consensus for the move (proposer plus two others in favour) that conformed with the stronger consensus on similar articles into an alleged lack of consensus. Hans Adler 11:01, 13 May 2011 (UTC)
 * The original move discussion was not closed prematurely, it was closed a day late. This is completely standard; there is no extra requirement that the discussion need be "settled" (whatever that is supposed to mean) – and even if there were, the close came four days after the last comment. I have no preference on the page title; I closed it as no consensus because there were two editors with reasonable arguments on either side (the "apple pie" rationale was not amongst them), a nomination in favour and no consensus on the broader issue elsewhere. A closure with any other result at any later time would have been negligent.  Skomorokh   13:10, 13 May 2011 (UTC)
 * Apparently I am not sufficiently familiar with RM culture. Merge discussions are sometimes open for years when there is little participation, and it's not clear to me why there is a pressing need to close a requested move with a weak move consensus as no consensus. (Probably some purely process-oriented reason that doesn't interest me.) On the other hand, as it can apparently be restarted with little fanfare, it doesn't do much harm, either. Hans Adler 16:02, 13 May 2011 (UTC)
 * That's about the sum of it, aye.  Skomorokh   16:13, 13 May 2011 (UTC)
 * Comment. This is not the first time that this editor has misrepresented the results of a Google search to attempt to prove the prevalence of "pi" over "&pi;" (and I and other editors have repeatedly called him on this).  If you look at the actual papers listed in that search, at least half of them use the symbol "&pi;" in their titles.  This is reminiscent of the post at Talk:Liu Hui's &pi; algorithm that purported to show that, because of the preponderance of hits for "Pi algorithm" at Google books, it means that most books spell out "pi" rather than using the symbol.  But in that case, the "Power Inversion algorithm" has nothing to do with &pi;!  The lesson in all of this is: Google searches are extremely poor indicators of the relative prevalence of "pi" versus "&pi;" in reliable sources, and so have very little value in this discussion.   Sławomir Biały  (talk) 18:55, 13 May 2011 (UTC)
 * Support per arguments already made at Talk:Liu Hui's &pi; algorithm.  Sławomir Biały  (talk) 10:32, 13 May 2011 (UTC)
 * Support per recent discussion at Wikipedia talk:WPM. RobHar (talk) 12:53, 13 May 2011 (UTC)
 * Oppose they should all be renamed to "pi" per discussion at Talk:Pi where &pi; is read by screen readers as "p" whereas it should be read as "pi"; and before anyone says that's how the Greeks call it, this is not Greek, it is English, and it is not the Greek letter, it's the math term; it gives the incorrect term in English, so accessibility is reduced. 184.144.163.181 (talk) 04:06, 14 May 2011 (UTC)
 * Referring to such an academically prolific constant throughout an encyclopedia in two different ways may lead to confused readers - regardless of whether they're using a screen reader or not. The consensus seems to be to use the symbol π to refer to the mathematical constant, and per the previous statement we should be consistent in doing so. What possible difference does it make how a screen reader pronounces something? Certainly no more difference than it makes if a human reader pronounces something incorrectly as they read an article. The main article can clarify pronunciation issues if need be. Ben (talk) 12:03, 14 May 2011 (UTC)
 * The real issue here is that mathematics articles are almost completely inaccessible with screenreaders anyway, because of the formulas. And if a screenreader cannot even deal with a single Greek letter within English text, then the screenreader is broken anyway. It's not part of Wikipedia's mission to create workarounds for broken software. Hans Adler 12:25, 14 May 2011 (UTC)
 * That raises an interesting point. How does the screen reader deal with exponents or even simpler: ab = c? How does the screen reader deal with Heteronyms? I suspect there are a very large number of examples where screen readers do not pronounce things 'correctly'. So why are we picking on π? Ben (talk) 12:36, 14 May 2011 (UTC)
 * Comment. I don't happen to think that the screen-reader argument compelling in general.  It's one thing to insist that the title of the main article pi should be read correctly by screen readers, but this is a concession of content and style in favor of accessibility.  It's another thing altogether to demand that content and style should always be secondary to accessibility.  And anyway, it seems to me that this is a technical issue with screen readers that should be fixed rather than something that should determine our house style.   Sławomir Biały  (talk) 12:57, 16 May 2011 (UTC)
 * We have things like the "DISPLAYTITLE" template, how difficult would it be to have a "READTITLE" template (ie a template that could more generally be used to write out in words what is written to help out screen readers)? RobHar (talk) 13:57, 16 May 2011 (UTC)
 * I'm afraid it's too much trouble. DISPLAYTITLE is also not as straightforward as it should be, due to vandalism. Similarly, a PRONOUNCETITLE template which can explain that Raymond Luxury Yacht is pronounced "Throatwobbler Mangrove" could also be used for various kinds of mischief that would only be noticed by screenreader users. Hans Adler 10:38, 26 May 2011 (UTC)


 * Oppose I prefer titles using the Latin Alphabet. The Proffesor (talk) 23:40, 27 May 2011 (UTC)
 * The above discussion is preserved as an archive of a requested move. Please do not modify it. Subsequent comments should be made in a new section on this talk page. No further edits should be made to this section.

Article title (again)
recently boldly moved this article to Gregory's Series and changed the lead paragraph to remove the reference to Leibniz and call it Gregory's formula. "Gregory's series" (note lowercase second word per Wikipedia style conventions) is certainly one of the names of this formula, but "Gregory's formula" or the "Gregory formula" appears to more commonly refer to something else involving numerical integration. In any case I reverted the change feeling that it was the sort of thing we should discuss here first. I have no strong feelings myself on which name we give the article as long as it is one of the common names for this series and is properly capitalized. Any opinions on this besides mine and OliverBel's? There are some discussions immediately above regarding the article title but they are on a different aspect of it (whether to spell out π or use the symbol for it). —David Eppstein (talk) 22:29, 17 May 2014 (UTC)

Sloppy proof
The proof, as written, is unacceptably sloppy - large leaps are made without justification, and a new variable (n) is introduced without explanation partway through (leaving the reader to guess as to what it actually is - an arbitrary integer? it is not clear).

Could someone please clean this up? 108.2.128.112 (talk) 03:31, 14 July 2018 (UTC)

Series is a rational number
The series is a sum of rational numbers, which can only result in a rational number. But π is proven to be an irrational number. It's apparent that the series therefore cannot EQUAL π (or π/4) since a rational number cannot equal an irrational number. The series must approach π but cannot equal π

2600:1003:B853:C0F:9512:7625:7F74:ED31 (talk) 02:49, 6 March 2021 (UTC)
 * What makes you think that the sum of an infinite series of rational numbers must be rational? —David Eppstein (talk) 02:56, 6 March 2021 (UTC)

The series can be expressed as this infinite loop, showing it's always the ratio of two integers:  N=1 S=4 Numerator=4 Denominator=1

Do print "π≈";Numerator;"/"; Denominator S=-S N=N+2 Numerator=N*Numerator+S*Denominator Denominator=Denominator*N Loop  2600:1003:B853:C0F:9512:7625:7F74:ED31 (talk) 03:21, 6 March 2021 (UTC)
 * It is rational at each finite step, but what makes you think that the result of an infinite process will keep the same properties as its finite steps? In any case this is far off-topic here, because this discussion board is only for discussing improvements to the article based on published reliable sources, not about clearing up your misunderstanding of basic mathematical definitions. —David Eppstein (talk) 04:59, 6 March 2021 (UTC)

The process can be repeated ad infinitum and the result can never be anything but a ratio of two integers. That much seems clear. This seems to be fundamentally different than something like (n/∞)=0 due to the difference between the rational series and it's irrational limit- which seems to perhaps impose another condition on determining equality.

What I question on the wiki page is the use of the "=" sign between the series and the "π" symbol. 2600:1003:B853:C0F:9512:7625:7F74:ED31 (talk) 05:52, 6 March 2021 (UTC)


 * Series is a limit of partial sums, IT IS NOT A SUM. Just by definition. There is no such concept as infinite sum, at least until summation rules are introduced, like Cesaro summation. 2A00:1FA0:48BD:B662:FD9A:5A04:FE1B:E20D (talk) 12:04, 4 April 2021 (UTC)