calculus - Find the value of sum (n/2^n)

I have the series $\sum_{n=0}^\infty \frac{n}{2^n}$. I must show that it converges to 2.

I was given a hint to take the derivative of $\sum_{n=0}^\infty x^n$ and multiply by $x$ , which gives

$\sum_{n=1}^\infty nx^n$ , or $\sum_{n=0}^\infty nx^n$.

Clearly if I take $x=\frac{1}{2}$ , the series is $\sum_{n=0}^\infty \frac{n}{2^n}$. How do I proceed from here?

Answer

Notice that if $|x|<1$ then the original series converges with

$$
\sum_{n=0}^\infty x^n \;\; =\;\; \frac{1}{1-x}.
$$

Computing the derivative and plugging in $x=\frac{1}{2}$ should hopefully seem easier now.