View source for Leibniz formula for π

{{Short description|Signed odd unit fractions sum to π/4}}
{{DISPLAYTITLE: Leibniz formula for {{pi}}}}
{{Pi box}}
{{for|other formulas known under the same name|List of things named after Gottfried Leibniz}}
In [[mathematics]], the '''Leibniz formula for [[Pi|{{pi}}]]''', named after [[Gottfried Wilhelm Leibniz]], states that
<math display=block>
\frac{\pi}{4} = 1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\frac{1}{9}-\cdots = \sum_{k=0}^{\infty} \frac{(-1)^{k}}{2k + 1},</math>
an [[alternating series]].
It is sometimes called the '''Madhava–Leibniz''' series as it was first discovered by the Indian mathematician [[Madhava of Sangamagrama]] or his followers in the 14th–15th century (see [[Madhava series]]),<ref>{{citation | last = Plofker | first = Kim | author-link = Kim Plofker | date = November 2012 | doi = 10.1007/s00283-012-9344-6 | issue = 1 | journal = [[The Mathematical Intelligencer]] | pages = 86–88 | title = ''Tantrasaṅgraha of Nīlakaṇṭha Somayājī'' by K. Ramasubramanian and M. S. Sriram | volume = 35| s2cid = 124507583 }}</ref> and was later independently rediscovered by [[James Gregory (mathematician)|James Gregory]] in 1671 and Leibniz in 1673.<ref>{{cite journal |last=Roy |first=Ranjan |year=1990 |title=The Discovery of the Series Formula for {{mvar|π}} by Leibniz, Gregory and Nilakantha |journal=Mathematics Magazine |volume=63 |number=5 |pages=291–306 |url=https://www.maa.org/sites/default/files/images/upload_library/22/Allendoerfer/1991/0025570x.di021167.02p0073q.pdf |doi=10.1080/0025570X.1990.11977541}}{{pb}}{{cite journal |last=Horvath |first=Miklos |title=On the Leibnizian quadrature of the circle. |journal=Annales Universitatis Scientiarum Budapestiensis (Sectio Computatorica) |volume=4 |year=1983 |pages=75–83 |url=http://ac.inf.elte.hu/Vol_004_1983/075.pdf }}</ref> The [[Taylor series]] for the [[inverse tangent]] function, often called ''[[Gregory's series]]'', is
<math display=block>
\arctan x = x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \cdots
= \sum_{k=0}^\infty \frac{(-1)^kx^{2k+1}}{2k+1}.
</math>
The Leibniz formula is the special case <math display=inline>\arctan 1 = \tfrac14\pi.</math><ref>{{citation|title=Special Functions|first1=George E. |last1=Andrews |first2=Richard |last2=Askey|first3=Ranjan |last3=Roy|publisher=[[Cambridge University Press]]|year=1999|isbn=0-521-78988-5|page=58}}</ref>
It also is the Dirichlet {{mvar|L}}-series of the non-principal [[Dirichlet character#Modulus 4|Dirichlet character]] of modulus&nbsp;4 evaluated at <math>s=1,</math> and therefore the value {{math|''β''(1)}} of the [[Dirichlet beta function]].
==Proofs==
=== Proof 1 ===
<math display=block>\begin{align}
\frac{\pi}{4} &= \arctan(1) \\ &= \int_0^1 \frac 1{1+x^2} \, dx \\[8pt]
& = \int_0^1\left(\sum_{k=0}^n (-1)^k x^{2k}+\frac{(-1)^{n+1}\,x^{2n+2} }{1+x^2}\right) \, dx \\[8pt]
& = \left(\sum_{k=0}^n \frac{(-1)^k}{2k+1}\right)
+(-1)^{n+1} \left(\int_0^1\frac{x^{2n+2}}{1+x^2} \, dx\right)
\end{align}</math>
Considering only the integral in the last term, we have:
<math display=block>0 \le \int_0^1 \frac{x^{2n+2}}{1+x^2}\,dx \le \int_0^1 x^{2n+2}\,dx = \frac{1}{2n+3} \;\rightarrow 0 \text{ as } n \rightarrow \infty.</math>
Therefore, by the [[squeeze theorem]], as {{math|''n'' → ∞}}, we are left with the Leibniz series:
<math display=block>\frac{\pi}4 = \sum_{k=0}^\infty\frac{(-1)^k}{2k+1}</math>
=== Proof 2 ===
Let <math>f(z) = \sum_{n=0}^{\infty}\frac{(-1)^n}{2n+1}z^{2n+1}</math>, when <math>|z|<1</math>, the series <math> \sum_{k=0}^\infty (-1)^k z^{2k}</math> converges uniformly, then
<math display=block>\arctan(z) = \int_{0}^{z} \frac {1}{1+t^2} dt =\sum_{n=0}^{\infty}\frac{(-1)^n}{2n+1}z^{2n+1} = f(z) \ (|z|<1). </math>
Therefore, if <math>f(z)</math> approaches <math>f(1)</math> so that it is continuous and converges uniformly, the proof is complete, where, the series <math>\sum_{n=0}^{\infty}\frac{(-1)^n}{2n+1}</math> to be converges by the [[Leibniz's test]], and also, <math>f(z)</math> approaches <math>f(1)</math> from within the Stolz angle, so from [[Abel's theorem]] this is correct.
== Convergence ==
{{comparison_pi_infinite_series.svg|400px|the Leibniz formula ({{cyan|''□''}}) and}}
Leibniz's formula converges extremely slowly: it exhibits [[sublinear convergence]]. Calculating {{pi}} to 10 correct decimal places using direct summation of the series requires precisely five billion terms because {{math|{{sfrac|4|2''k'' + 1}} < 10<sup>−10</sup>}} for {{math|''k'' > 2 × 10<sup>10</sup> − {{sfrac|1|2}}}} (one needs to apply [[Alternating series#Approximating sums|Calabrese error bound]]). To get 4 correct decimal places (error of 0.00005) one needs 5000 terms.<ref>{{Cite journal |last=Villarino |first=Mark B. |date=2018-04-21 |title=The Error in an Alternating Series |url=https://doi.org/10.1080/00029890.2017.1416875 |journal=The American Mathematical Monthly |volume=125 |issue=4 |pages=360–364 |doi=10.1080/00029890.2017.1416875 |hdl=10669/75532 |s2cid=56124579 |issn=0002-9890|hdl-access=free }}</ref> Even better than Calabrese or Johnsonbaugh error bounds are available.<ref>{{Cite arXiv|last=Rattaggi |first=Diego |date=2018-08-30 |title=Error estimates for the Gregory-Leibniz series and the alternating harmonic series using Dalzell integrals |class=math.CA |eprint=1809.00998}}</ref>
However, the Leibniz formula can be used to calculate {{pi}} to high precision (hundreds of digits or more) using various [[convergence acceleration]] techniques. For example, the [[Shanks transformation]], [[Euler transform]] or [[Van Wijngaarden transformation]], which are general methods for alternating series, can be applied effectively to the partial sums of the Leibniz series. Further, combining terms pairwise gives the non-alternating series
<math display=block>\frac{\pi}{4} = \sum_{n=0}^{\infty} \left(\frac{1}{4n+1}-\frac{1}{4n+3}\right) = \sum_{n=0}^{\infty} \frac{2}{(4n+1)(4n+3)}</math>
which can be evaluated to high precision from a small number of terms using [[Richardson extrapolation]] or the [[Euler–Maclaurin formula]]. This series can also be transformed into an integral by means of the [[Abel–Plana formula]] and evaluated using techniques for [[numerical integration]].
== Unusual behaviour ==
If the series is truncated at the right time, the [[Decimal representation|decimal expansion]] of the approximation will agree with that of {{pi}} for many more digits, except for isolated digits or digit groups. For example, taking five million terms yields
<math display=block>3.141592\underline{4}5358979323846\underline{4}643383279502\underline{7}841971693993\underline{873}058...</math>
where the underlined digits are wrong. The errors can in fact be predicted; they are generated by the [[Euler number]]s {{math|''E<sub>n</sub>''}} according to the [[asymptotic expansion|asymptotic]] formula
<math display=block>\frac{\pi}{2} - 2 \sum_{k=1}^{N/2} \frac{(-1)^{k-1}}{2k-1} \sim \sum_{m=0}^\infty \frac{E_{2m}}{N^{2m+1}}</math>
where {{math|''N''}} is an integer divisible by 4. If {{math|''N''}} is chosen to be a power of ten, each term in the right sum becomes a finite decimal fraction. The formula is a special case of the [[Euler–Boole summation]] formula for alternating series, providing yet another example of a convergence acceleration technique that can be applied to the Leibniz series. In 1992, [[Jonathan Borwein]] and Mark Limber used the first thousand Euler numbers to calculate {{pi}} to 5,263 decimal places with the Leibniz formula.<ref>{{citation|last1=Borwein|first1=Jonathan|author1-link=Jonathan Borwein|last2=Bailey|first2=David|author2-link=David H. Bailey (mathematician)|last3=Girgensohn|first3=Roland|contribution=1.8.1: Gregory's Series Reexamined|contribution-url=https://books.google.com/books?id=10BZDwAAQBAJ&pg=PA28|isbn=1-56881-136-5|mr=2051473|pages=28–30|publisher=A K Peters|title=Experimentation in mathematics: Computational paths to discovery|year=2004}}</ref>
==Euler product==
The Leibniz formula can be interpreted as a [[Dirichlet series]] using the unique non-principal [[Dirichlet character]] modulo&nbsp;4. As with other Dirichlet series, this allows the infinite sum to be converted to an [[infinite product]] with one term for each [[prime number]]. Such a product is called an [[Euler product]]. It is:
<math display=block>\begin{align}
\frac\pi4 &= \biggl(\prod_{p \,\equiv\, 1\ (\text{mod}\ 4)}\frac{p}{p-1}\biggr)
\biggl( \prod_{p\,\equiv\, 3\ (\text{mod}\ 4)}\frac{p}{p+1}\biggr) \\[7mu]
&= \frac{3}{4} \cdot \frac{5}{4} \cdot \frac{7}{8} \cdot
\frac{11}{12} \cdot \frac{13}{12} \cdot \frac{17}{16} \cdot
\frac{19}{20} \cdot \frac{23}{24} \cdot \frac{29}{28} \cdots
\end{align}</math>
In this product, each term is a [[superparticular number|superparticular ratio]], each numerator is an odd prime number, and each denominator is the nearest multiple of 4 to the numerator.<ref>{{citation|title=The Legacy of Leonhard Euler: A Tricentennial Tribute|first=Lokenath|last=Debnath|publisher=World Scientific|year=2010|isbn=9781848165267|page=214|url=https://books.google.com/books?id=K2liU-SHl6EC&pg=PA214}}.</ref> The product is conditionally convergent; its terms must be taken in order of increasing {{mvar|p}}.
==See also==
*[[List of formulae involving π|List of formulae involving {{pi}}]]
==References==
{{reflist}}
{{DEFAULTSORT:Leibniz formula for Pi}}
[[Category:Pi algorithms]]
[[Category:Articles containing proofs]]
[[Category:Gottfried Wilhelm Leibniz]]
[[Category:Mathematical series]]