Finding $\sum\limits_{i=0}^{\infty}\frac{i}{4^i}$

Question

I'm struggling with finding this sum:

$$\sum\limits_{i=0}^{\infty}\frac{i}{4^i}$$

Any pointers would be greatly appreciated.

I have not found any information of use elsewhere, but maybe I am using the wrong search terms.

Answer 1

Let $S=\sum\limits_{i=0}^{\infty}\frac{i}{4^i}$, so $4S=\sum\limits_{i=1}^{\infty}\frac{i}{4^{i-1}}=\sum\limits_{i=0}^{\infty}\frac{i+1}{4^i}$, so:

$3S=4S-S=\sum\limits_{i=0}^{\infty}\frac{1}{4^i}=\frac{1}{1-\frac{1}{4}}=\frac{4}{3}$

$S=\frac{4}{9}$