TITLE: Limit of distance between two random points in a unit-radius $n$-sphere
QUESTION [12 upvotes]: 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!

REPLY [15 votes]: Both points will be very close to (let's pretend: on) the surface with prob almost 1. Call the first point the north pole. By concentration of measure for the sphere, a randomly chosen second point is almost guaranteed to be almost on the equator, so the limit should be $\sqrt{2}$.