Basic asymptotic estimates for powers of Wallis’ ratios

  • Vito Lampret University of Ljubljana, Ljubljana, 386 Slovenia, EU.
Keywords: approximation, asymptotic, estimate, inequality, power, Wallis’ ratio

Abstract

For any \(a\in{\mathbb R}\), for every \(n\in{\mathbb N}\), 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.

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Published
2021-12-01
How to Cite
[1]
V. Lampret, “Basic asymptotic estimates for powers of Wallis’ ratios”, CUBO, vol. 23, no. 3, pp. 357–368, Dec. 2021.
Section
Articles