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# The Best Bounds in Wallis' Inequality

Chao-Ping Chen and Feng Qi
Proceedings of the American Mathematical Society
Vol. 133, No. 2 (Feb., 2005), pp. 397-401
Stable URL: http://www.jstor.org/stable/4097942
Page Count: 5
For all natural numbers n, let n!! denote a double factorial. Then $\frac{1}{\sqrt{\pi (n + \frac{4}{\pi} - 1)}} \leq \frac{(2n - 1)!!}{(2n)!!} < \frac{1}{\sqrt{\pi (n + \frac{1}{4})}}$ The constants $\frac{4}{\pi} - 1$ and 1/4 are the best possible. From this, the well-known Wallis' inequality is improved.