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{{DISPLAYTITLE: Leibniz formula for {{pi}}}}
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:''See [[List of things named after Gottfried Leibniz]] for other formulas known under the same name.''

In [[mathematics]], the '''Leibniz formula for [[Pi|{{pi}}]]''', named after [[Gottfried Leibniz]], states that

:<math>1 \,-\, \frac{1}{3} \,+\, \frac{1}{5} \,-\, \frac{1}{7} \,+\, \frac{1}{9} \,-\, \cdots \,=\, \frac{\pi}{4}.</math>

It is also called the '''Madhava–Leibniz''' series as it is a [[special case]] of a more general series expansion for the [[inverse tangent]] function, first discovered by the Indian mathematician [[Madhava of Sangamagrama]] in the 14th century, the specific case first published by Leibniz around 1676.<ref>{{Cite book|last=Charles Henry Edwards|title=The historical development of the calculus|publisher=Springer|year=1994|edition=3|series=Springer Study Edition Series|page=247|isbn=978-0-387-94313-8}}</ref> The series for the [[inverse tangent]] function, which is also known as [[Gregory's series]], can be given by:
: <math>\arctan x = x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \cdots </math>

The Leibniz formula for {{sfrac|{{pi}}|4}} can be obtained by putting {{math|''x'' {{=}} 1}} into this series.<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 4 evaluated at {{math|''s'' {{=}} 1}}, and therefore the value {{math|''β''(1)}} of the [[Dirichlet beta function]].

==Proof==
:<math>\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 line, we have:

:<math>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>\frac{\pi}4 = \sum_{k=0}^\infty\frac{(-1)^k}{2k+1}</math>

== 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 about five billion terms because {{math|{{sfrac|1|2''k'' + 1}} < 10<sup>−10</sup>}} for {{math|''k'' > 5 × 10<sup>9</sup> − {{sfrac|1|2}}}}.

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>\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 behavior ==
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

:3.141592<u>4</u>5358979323846<u>4</u>643383279502<u>7</u>841971693993<u>873</u>058...

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>\frac{\pi}{2} - 2 \sum_{k=1}^\frac{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 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.

==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>\frac\pi4=\left(\prod_{p\equiv 1\pmod 4}\frac{p}{p-1}\right) \left( \prod_{p\equiv 3\pmod 4}\frac{p}{p+1}\right)=\frac{3}{4} \cdot \frac{5}{4} \cdot \frac{7}{8} \cdot \frac{11}{12} \cdot \frac{13}{12}\cdot\frac{17}{16} \cdots</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>

==See also==
*[[List of formulae involving π|List of formulae involving {{pi}}]]

==Notes==
{{reflist}}

==References==
* Jonathan Borwein, David Bailey & Roland Girgensohn, ''Experimentation in Mathematics - Computational Paths to Discovery'', A K Peters 2003, {{isbn|1-56881-136-5}}, pages 28&ndash;30.

==External links==
* [http://mattst88.com/programming/?page=leibniz Leibniz Formula in C, x86 FPU Assembly, x86-64 SSE3 Assembly, and DEC Alpha Assembly]

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[[Category:Pi algorithms]]
[[Category:Articles containing proofs]]
[[Category:Gottfried Leibniz]]