real analysis - Computing $sum_{ngeq 0}nfrac{1}{4^n}$

Can I compute the sum
$$
\sum_{n\geq 0}n\frac{1}{4^n}

$$
by use of some trick?

First I thought of a geometrical series?

Answer

This is very similar to the geometric series, infact $$\sum_{n=0}^\infty n x^n = x \cdot \frac{d}{dx} \sum_{n=0}^\infty x^n = x \cdot \frac{d}{dx} \frac{1}{1-x} = \frac{x}{(1-x)^2}$$

Plugging in $x=1/4$ will provide you with an answer.