Basic asymptotic estimates for powers of Wallis’ ratios

Abstract

For any $a\in\R$, for every $n\in\N$, and for $n$-th Wallis' ratio $w_n:=\prod_{k=1}^n\frac{2k-1}{2k}$, the relative error $r_{\,\!_0}(a,n):=\big(v_{\,\!_0}(a,n)-w_n^a\big)/w_n^a$ of the approximation $w_n^a\approx v_{\,\!_0}(a,n):=(\pi n)^{-a/2} $ is estimated as $ \big|r_{\,\!_0}(a,n)\big| < \frac{1}{4n}$. The improvement $w_n^a\approx v(a,n):=(\pi n)^{-a/2}\left(1-\frac{a}{8n} +\frac{a^2}{128n^2}\right)$ is also studied.

Keywords

• approximation
• asymptotic
• estimate
• inequality
• power
• wallis’ ratio