Leibniz formula for π

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the mathematical constant π
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Squaring the circle Basel problem Tau (τ) Other topics related to π

See Leibniz (disambiguation) for other formulas known under the same name.

In mathematics, the Leibniz formula for π, named after Gottfried Leibniz, states that

$1 \,-\, \frac{1}{3} \,+\, \frac{1}{5} \,-\, \frac{1}{7} \,+\, \frac{1}{9} \,-\, \cdots \;=\; \frac{\pi}{4}.\!$

Using summation notation:

$\sum_{n=0}^\infty \, \frac{(-1)^n}{2n+1} \;=\; \frac{\pi}{4}.\!$

Names

The infinite series above is called the Leibniz series. It is also called the Gregory–Leibniz series, recognizing the work of James Gregory. The formula was discovered by Madhava of Sangamagrama and so is also called the Madhava–Leibniz series.

It also is the Dirichlet L-series of the non-principal Dirichlet character of modulus 4 evaluated at s=1, and therefore the value β(1) of the Dirichlet beta function.

Proof

$\begin{align} \frac{\pi}{4} & = \arctan(1)\;=\;\int_0^1 \frac 1{1+x^2} \, dx \\[8pt] & = \int_0^1\left(\sum_{k=0}^n (-1)^k x^{2k}+\frac{(-1)^{n+1}\,x^{2n+2} }{1+x^2}\right) \, dx \\[8pt] & = \sum_{k=0}^n \frac{(-1)^k}{2k+1} +(-1)^{n+1}\int_0^1\frac{x^{2n+2}}{1+x^2} \, dx. \end{align}$

Considering only the integral in the last line, we have:

$0 < \int_0^1 \frac{x^{2n+2}}{1+x^2}\,dx \;<\; \int_0^1 x^{2n+2}\,dx \;=\; \frac{1}{2n+3} \;\rightarrow\; 0 \text{ as } n \rightarrow \infty.\!$

Therefore, as n → ∞ we are left with the Leibniz series:

$\frac{\pi}4\;=\;\sum_{k=0}^\infty\frac{(-1)^k}{2k+1}.$

Inefficiency

Leibniz's formula converges slowly. Calculating π to 10 correct decimal places using direct summation of the series requires about 5,000,000,000 terms because $\scriptstyle \frac 1{2k+1}<10^{-10}$ for $\scriptstyle k>\frac{10^{10}-1}2$.

However, the Leibniz formula can be used to calculate π 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

$\frac{\pi}{4} = \sum_{n=0}^{\infty} \bigg(\frac{1}{4n+1}-\frac{1}{4n+3}\bigg) = \sum_{n=0}^{\infty} \frac{2}{(4n+1)(4n+3)}$

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 expansion of the approximation will agree with that of π for many more digits, except for isolated digits or digit groups. For example, taking 5,000,000 terms yields

3.1415924535897932384646433832795027841971693993873058...

where the underlined digits are wrong. The errors can in fact be predicted; they are generated by the Euler numbers E_n according to the asymptotic formula

$\frac{\pi}{2} - 2 \sum_{k=1}^{N/2} \frac{(-1)^{k-1}}{2k-1} \sim \sum_{m=0}^{\infty} \frac{E_{2m}}{N^{2m+1}}\!$

where N is an integer divisible by 4. If 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 π to 5,263 decimal places with Leibniz' formula.

References

• Jonathan Borwein, David Bailey & Roland Girgensohn, Experimentation in Mathematics - Computational Paths to Discovery, A K Peters 2003, ISBN 1-56881-136-5, pages 28–30.

External links

• Leibniz Formula in C, x86 FPU Assembly, x86-64 SSE3 Assembly, and DEC Alpha Assembly