An asymptotic approximation of Wallis’ sequence

Vito Lampret

Open Mathematics, Tome 10 (2012), p. 775-787 / Harvested from The Polish Digital Mathematics Library

Résumé

An asymptotic approximation of Wallis’ sequence W(n) = Πk=1n 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) = Πk=1n(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, $W(n) \cdot (a_n + b_n ) < \pi < W(n) \cdot (a_n + b^{\prime }_n )$ with $a_n = 2 + \frac{1}{{2n + 1}} + \frac{2}{{3(2n + 1)^2 }} - \frac{1}{{3n(2n + 1)^{\prime }}}b_n = \frac{2}{{33(n + 1)^{2^{\prime }} }}b^{\prime }_n \frac{1}{{13n^{2^{\prime }} }}n \in \mathbb {N} $ .

Détail

Publié le : 2012-01-01

Zbl 1252.40002

EUDML-ID : urn:eudml:doc:269161

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