This is a companion contrast to the earlier analogous question for unit $n$-cubes, where the answer (provided by several respondents) is $\infty$ .

What is the limit, as $n \to \infty$, of the expected distance between two points chosen uniformly at random within a unit-radius hypersphere in $\mathbb{R}^n$, i.e., in the unit-radius $n$-ball?

Dividing OEIS A093530 by A093531 I see that it appears to be approaching approximately $1.37$ for odd $n$, but I wonder if the limit is actually known, either exactly or to significant precision? I cannot quite extract an answer from the MathWorld article...

This certainly provides a dramatic contrast between the $n$-cube and the $n$-sphere!