Sequence sum question: $\sum_{n=0}^{\infty}nk^n$ I am very confused about how to compute 
$$\sum_{n=0}^{\infty}nk^n.$$
Can anybody help me?

Asssuming $|k|<1$, this series converges uniformly and therefore sum and derivative can be interchanged:
$\sum_{n=0}^{\infty}nk^n=k \sum_{n=0}nk^{n-1} = k \sum_{n=0}^{\infty}\frac{d}{dk}(k^n)
=k \frac{d}{dk} \sum_{n=0}^{\infty}k^n = k \frac{d}{dk}\frac{1}{1-k}=\frac{k}{(1-k)^2}$
EDIT: in case $k>1$ this is a completely different situation, as the geometric series diverges, but I don't know how to solve this problem.
