Approximation and bounds for the Wallis ratio

TL;DR

This paper introduces an improved continued fraction approximation for the Wallis ratio, offering faster computation and tighter bounds compared to existing methods, supported by numerical evidence.

Contribution

The paper presents a novel continued fraction approximation for the Wallis ratio along with double-sided inequalities, enhancing computational efficiency and accuracy.

Findings

The new approximation outperforms recent asymptotic series in speed.

Numerical results demonstrate the superiority of the proposed approximation.

Double-sided inequalities provide bounds for the Wallis ratio.

Abstract

In this paper, we present an improved continued fraction approximation of the Wallis ratio. This approximation is fast in comparison with the recently discovered asymptotic series. We also establish the double-side inequality related to this approximation. Finally, some numerical computations are provided for demonstrating the superiority of our approximation.