An asymptotic approximation of Wallis’ sequence

Abstract

An asymptotic approximation of Wallis’ sequence W(n) = Π _k=1 ⁿ 4k ²/(4k ² − 1) obtained on the base of Stirling’s factorial formula is presented. As a consequence, several accurate new estimates of Wallis’ ratios w(n) = Π _k=1 ⁿ (2k−1)/(2k) are given. Also, an asymptotic approximation of π in terms of Wallis’ sequence W(n) is obtained, together with several double inequalities such as, for example, $$W(n) \cdot (a_n + b_n ) < \pi < W(n) \cdot (a_n + b'_n )$$ with $$a_n = 2 + \frac{1} {{2n + 1}} + \frac{2} {{3(2n + 1)^2 }} - \frac{1} {{3n(2n + 1)'}}b_n = \frac{2} {{33(n + 1)^{2'} }}b'_n \frac{1} {{13n^{2'} }}n \in \mathbb{N} $$ .