How to derive the closed form of the sum of $kr^k$ $$
\sum_{k=0}^{n}kr^k = r\frac{1-(n+1)r^n + nr^{n+1}}{ (1 - r)^2 }
$$
How to derive it? I read about some finite calculus, and i understand how to tackle sums of $x^2$, $x^3$, etc.. But I don't know if the same methods can be used on this sum? 

Hint: $kr^k = r k r^{k-1} = r \frac{d}{dr}r^{k}$, now interchange summation and derivative.  
