New Bounds and Asymptotic Expansions for the Volume of the Unit Ball in Rn Based on Pade Approximation

Let Omega(n )= pi(n/2)/Gamma(n/2 + 1)(n is an element of N) denote the volume of the unit ball in R-n. In this paper, we present asymptotic expansions and new bound related to the quantities: Omega(n)/Omega(n-1) and Omega(2)(n)/Omega(n-1)Omega(n+1). Finally, a numerical example is provided to demonstrate the effectiveness and feasibility of the proposed method.

Automated summary

This paper discusses the Omega function and related quantities, and presents asymptotic expansions and new bounds. The effectiveness and feasibility of the proposed method are demonstrated through a numerical example.