I was playing around with the volume of n-dimensional spheres to get a better understanding of the "curse of dimensionality" and I realized that the volume of a n-dimensional sphere tends to 0 for a given radius and $n \to \infty$ Also it increases in the beginning but starts to decrease at some point. Is there an intiution why this happens?
Recall that the volume of a n-dimensional sphere with radius $r$ is$\frac{\pi^{0.5n}}{\Gamma(0.5n+1)}r^n$