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| {{DISPLAYTITLE: Leibniz formula for {{pi}}}}
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| {{Pi box}}
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| :''See [[Leibniz (disambiguation)]] for other formulas known under the same name.''
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| In [[mathematics]], the '''Leibniz formula for [[Pi|{{pi}}]]''', named after [[Gottfried Leibniz]], states that
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| :<math>1 \,-\, \frac{1}{3} \,+\, \frac{1}{5} \,-\, \frac{1}{7} \,+\, \frac{1}{9} \,-\, \cdots \;=\; \frac{\pi}{4}.\!</math>
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| Using [[summation]] notation:
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| :<math>\sum_{n=0}^\infty \, \frac{(-1)^n}{2n+1} \;=\; \frac{\pi}{4}.\!</math>
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| ==Names==
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| {{See also|Madhava of Sangamagrama}}
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| The [[infinite series]] above is called the '''Leibniz series'''. It is also called the '''[[Gregory's series|Gregory–Leibniz series]]''', recognizing the work of [[James Gregory (mathematician)|James Gregory]]. The formula was ''first'' discovered by [[Madhava of Sangamagrama]]<ref>{{citation|title=Special Functions|last=George E. Andrews, Richard Askey|first=Ranjan Roy|publisher=[[Cambridge University Press]]|year=1999|isbn=0-521-78988-5|page=58}}</ref> in the 14th century, but was not widely known in the West. Since Leibniz, however, was the first person to rediscover the series in continental Europe, it is named after him. In retrospect, the naming is sometimes modified to '''[[Madhava series|Madhava–Leibniz series]]''' in order to recognize Madhava's contribution.<ref>{{citation|first=R. C.|last=Gupta|title=On the remainder term in the Madhava–Leibniz's series|journal=Ganita Bharati|volume=14|issue=1-4|year=1992|pages=68–71}}</ref>
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| It also is the Dirichlet L-series of the non-principal [[Dirichlet character#Modulus 4|Dirichlet character]]
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| of modulus 4 evaluated at {{math|s{{=}}1}}, and therefore the value {{math|β(1)}} of the [[Dirichlet beta function]].
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| ==Proof==
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| :<math>
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| \begin{align}
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| \frac{\pi}{4} & = \arctan(1)\;=\;\int_0^1 \frac 1{1+x^2} \, dx \\[8pt]
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| & = \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]
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| & = \sum_{k=0}^n \frac{(-1)^k}{2k+1}
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| +(-1)^{n+1}\int_0^1\frac{x^{2n+2}}{1+x^2} \, dx.
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| \end{align}
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| </math>
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| Considering only the integral in the last line, we have:
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| :<math>0 < \int_0^1 \frac{x^{2n+2}}{1+x^2}\,dx \;<\; \int_0^1 x^{2n+2}\,dx \;=\; \frac{1}{2n+3} \;\rightarrow\; 0 \text{ as } n \rightarrow \infty.\!</math>
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| Therefore, as {{math|''n'' → ∞}} we are left with the Leibniz series:
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| :<math>\frac{\pi}4\;=\;\sum_{k=0}^\infty\frac{(-1)^k}{2k+1}.</math>
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| == Inefficiency ==
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| Leibniz's formula converges slowly. Calculating {{pi}} to 10 ''correct'' decimal places using direct summation of the series requires about 5,000,000,000 terms because <math>\scriptstyle \frac 1{2k+1}<10^{-10}</math> for <math>\scriptstyle k>\frac{10^{10}-1}2</math>.
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| 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
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| :<math>\frac{\pi}{4} = \sum_{n=0}^{\infty} \bigg(\frac{1}{4n+1}-\frac{1}{4n+3}\bigg) = \sum_{n=0}^{\infty} \frac{2}{(4n+1)(4n+3)}</math>
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| 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]].
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| == Unusual behavior ==
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| 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 5,000,000 terms yields
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| :3.141592<u>4</u>5358979323846<u>4</u>643383279502<u>7</u>841971693993<u>873</u>058...
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| 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 | |
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| :<math>\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> | |
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| 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 Leibniz' formula.
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| ==Euler product==
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| The Leibniz formula can be interpreted as a [[Dirichlet series]] using the (unique) [[Dirichlet character]] modulo 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:
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| :<math>\pi/4=\left(\prod_{p\equiv 1\pmod 4}\frac{p}{p-1}\right)\cdot\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>
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| In this product, each term is a [[superparticular number|superparticular ratio]], each numerator is a prime number, and each denominator is the nearest multiple of four 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=http://books.google.com/books?id=K2liU-SHl6EC&pg=PA214}}.</ref>
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| ==See also==
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| *[[Gregory's series|Gregory–Leibniz series]]
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| *[[Madhava of Sangamagrama]]
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| *[[Madhava series|Madhava–Leibniz series]]
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| ==Notes==
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| {{reflist}}
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| ==References==
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| * Jonathan Borwein, David Bailey & Roland Girgensohn, ''Experimentation in Mathematics - Computational Paths to Discovery'', A K Peters 2003, ISBN 1-56881-136-5, pages 28–30.
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| ==External links==
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| * [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]]
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| [[Category:Articles containing proofs]]
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| [[Category:Gottfried Leibniz]]
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Greetings! I am Marvella and I feel comfy when individuals use the full name. To collect cash is a factor that I'm totally addicted to. Managing people has been his working day job for a whilst. North Dakota is exactly where me and my spouse reside.
My weblog :: healthy meals delivered