I hate this result because it's not actually saying what it's advertized to be saying

sure, you already got the complaint about comparing values of different units - but observe HOW this question is actually sidestepped! We divide hyper-volume of n-sphere by hyper-volume of n-cube!

Now this raises the question: WHAT n-cube are we taking?

If you take hyper-cube with side-length of sphere's diameter, you'll have nice relation between cube and its inscribed sphere - and, predictably, as n goes up, number of cube's "corners" also goes up. So this ratio consistently goes down

But what about your numbers? Well that result happens when you take cube with side-length of sphere's RADIUS. That way you arbitrarily add a scaling factor 2^n - and there's nothing geometric about this behaviour