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.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Similar Papers
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.