I've read on a paper that, in the two dimensional case, if you start from the origin and take steps of length one in an arbitrary direction (uniformely on the unit sphere $S^1$), the expected distance after $n$ steps from the starting point is approximated by $\sqrt{n\pi}/2$.
(source: https://pdfs.semanticscholar.org/6685/1166d588821456477f2007a37bf0428a2cf2.pdf).
I was wondering if there was a similar formula for higher dimensional random walks, which means:
Starting from the origin in $\mathbb{R}^d$, if i set $v_i = v_{i-1} + \mu$, where $\mu \in S^{d-1}$ picked at random at every step and $v_0 = 0$, what is the expected value of $|v_n|$? e.g. how distant is the point from the origin after having taken $n$ steps in random directions?
I don't need a precise formula, everything that just gives an idea on how large the expected distance is works just fine. (if you could add a reference to the answer as well it would be great! :D )
Thank you very much!