Abstract
For all natural numbers n n , let n ! ! n!! denote a double factorial. Then 1 π ( n + 4 π − 1 ) ≤ ( 2 n − 1 ) ! ! ( 2 n ) ! ! > 1 π ( n + 1 4 ) . \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 4 π − 1 \frac {4}{\pi }-1 and 1 4 \frac 14 are the best possible. From this, the well-known Wallis’ inequality is improved.
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