# Wallis formula

A formula which expresses the number $\pi /2$ as an infinite product:

$$\tag{1 } { \frac \pi {2} } = \ \left ( { \frac{2}{1} } \cdot { \frac{2}{3} } \right ) \left ( { \frac{4}{3} } \cdot { \frac{4}{5} } \right ) \dots \left ( { \frac{2k}{2k-1} } \cdot { \frac{2k}{2k+1} } \right ) \dots =$$

$$= \ \lim\limits _ {m \rightarrow \infty } \prod _ { k=1 } ^ { m } \frac{( 2k) ^ {2} }{( 2k- 1)( 2k+ 1) } .$$

There exist other variants of this formula, e.g.:

$$\tag{2 } \sqrt \pi = \ \lim\limits _ {m \rightarrow \infty } \ \frac{( m!) ^ {2} \cdot 2 ^ {2m} }{( 2m)! \sqrt m } .$$

Formula (1) was first employed by J. Wallis  in his calculation of the area of a disc; it is one of the earliest examples of an infinite product.

How to Cite This Entry:
Wallis formula. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Wallis_formula&oldid=51104
This article was adapted from an original article by T.Yu. Popova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article