Question

Does anybody know a closed-form solution for the following expression (N>=1)? I don't even know where to begin with the i+n denominator.

Sum of i=0 to n Combin(n,i) * (2i/(i+n)) / (2^n)

Answer 1

$\sum_{i=0}^{n}2^{-n} {n\choose i}\frac{2i}{i+n}=2^{1-n}\frac{n}{1+n}F(1-n,1+n;2+n;-1)$

with $F$ the hypergeometric function.