sequences and series - Need to know why $sum_{k=0}^{infty}kr^{k} = frac{r}{(1-r)^{2}}$

Working on a Stat problem where I must find $E(x)$ of $f(x)=\left(\frac{1}{2}\right)^{x+1}$ for $x=0,1,2,\cdots$

I have,

$$E(x)=\sum_{x=0}^{\infty}x\left(\frac{1}{2}\right)^{x+1}=\frac{1}{2}\sum_{x=0}^{\infty}x\left(\frac{1}{2}\right)^{x}$$

I'm pretty sure this is a geometric series, but it would defeat the purpose of doing this problem if I didn't know why the following sum converges:

$$\sum_{k=0}^{\infty}kr^{k} = \frac{r}{(1-r)^{2}}$$

Can anyone explain why this is? I've tried using derivative but keep going in circles.