# Wallis formulae

Wallis’ formula expresses $\pi$ as an infinite product:

 $\frac{\pi}{2}=\prod_{n=1}^{\infty}\frac{4n^{2}}{4n^{2}-1}=\frac{2}{1}\frac{2}{% 3}\frac{4}{3}\frac{4}{5}\cdots$

It may be derived by taking the limit as $n\to\infty$ of the ratio of the following two integrals.

 $\int_{0}^{\frac{\pi}{2}}\sin^{2n}xdx=\frac{1\cdot 3\cdots(2n-1)}{2\cdot 4% \cdots 2n}\frac{\pi}{2}$
 $\int_{0}^{\frac{\pi}{2}}\sin^{2n+1}xdx=\frac{2\cdot 4\cdots 2n}{3\cdot 5\cdots% (2n+1)}$
Title Wallis formulae WallisFormulae 2013-03-22 12:56:15 2013-03-22 12:56:15 rspuzio (6075) rspuzio (6075) 8 rspuzio (6075) Definition msc 40A20 msc 40A10 Pi ReductionFormulasForIntegrationOfPowers