Hypercubes circumscribed in hyperspheres: a constant growth ratio for volumes at large dimensions

This note shows that an interesting property arises when considering the relation between the hypersphere volumes at dimensions $n+1$ and $n$, if the hyperspheres circumscribe unitary hypercubes in $n+1$ and $n$ dimensions, respectively . In the limit of large $n$, the ratio $\frac{V_{n+1}}{V_{n}}$ is constant and equal to the irrational number $\sqrt{\frac{\pi e}{2}}$. Some consequences of this result are explored.