For me the n-sphere problem makes intuitive sense if you think of it as a subset of the volume of a unit n-cube: No matter what n is, a unit n-cube has volume

For me the n-sphere problem makes intuitive sense if you think of it as a subset of the volume of a unit n-cube: No matter what n is, a unit n-cube has volume 1^n = 1. So the problem is equivalent to: what proportion of that 1 is inside the n-sphere? And then think about how the difference between a unit cube with a unit cylinder vs. a unit sphere in it: the cylinder takes the same proportion as a circle from a square; the sphere takes a subset of that because instead of just pushing the circle through the cube, you rotate it around a point. A rigorous proof would show that the series in fact trends to zero, and not just monotonically smaller values, but for intuitive purposes, that's good enough for me.