Wallis formulae

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

$$\frac{\pi }{2}=\prod _{n=1}^{\mathrm{\infty }}\frac{4n^2}{4n^2-1}=\frac{2}{1}\frac{2}{3}\frac{4}{3}\frac{4}{5}\mathrm{\cdots }$$

It may be derived by taking the limit as $n\to \mathrm{\infty }$ of the ratio of the followingtwo integrals.

$${\int }_0^{\frac{\pi }{2}}{\sin }^{2n}xdx=\frac{1\cdot 3\mathrm{\cdots }(2n-1)}{2\cdot 4\mathrm{\cdots }2n}\frac{\pi }{2}$$

$${\int }_0^{\frac{\pi }{2}}{\sin }^{2n+1}xdx=\frac{2\cdot 4\mathrm{\cdots }2n}{3\cdot 5\mathrm{\cdots }(2n+1)}$$