The best bounds in Wallis’ inequality

Chen, Chao-Ping; Qi, Feng

doi:10.1090/S0002-9939-04-07499-4

The best bounds in Wallis’ inequality
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by Chao-Ping Chen and Feng Qi PDF

Proc. Amer. Math. Soc. 133 (2005), 397-401 Request permission

Abstract:

For all natural numbers $n$, let $n!!$ denote a double factorial. Then \begin{equation*} \frac 1{\sqrt {\pi \bigl (n+\frac 4{\pi }-1\bigr )}}\leq \frac {(2n-1)!!}{(2n)!!}<\frac 1{\sqrt {\pi \bigl (n+\frac 14\bigr )}}. \end{equation*} The constants $\frac {4}{\pi }-1$ and $\frac 14$ are the best possible. From this, the well-known Wallis’ inequality is improved.

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