Unit ball in high dimensions

Abstract

In this article, we compute the volume V _n of the unit ball in an n-dimensional space. For n = 1, 2, 3, the volumes are respectively 2, π 4π /3, which are the length of interval [-1,1], area of a unit circle and volume of the unit sphere. The numbers V _n 'appear' to increase. But in fact this not so. In fact V _n tends to zero as n tends to infinity!.