sequences and series - Show that $sum_{n = 1}^{+infty} frac{n}{2^n} = 2$

Show that $\sum_{n = 1}^{+\infty} \frac{n}{2^n} = 2$. I have no idea to solve this problem. Anyone could help me?

Answer

Consider the following

$$\frac{1}{1-x} = \sum_{k\geq 0}x^ k $$

The series converges for $|x|<1$

Differentiating both sides we have

$$\frac{1}{(1-x)^2} = \sum_{k\geq 1}k x^{k-1} $$

$$\frac{x}{(1-x)^2} = \sum_{k\geq 1}k x^{k} $$

Now put $x=\frac{1}{2}$

$$2 = \sum_{k\geq 1}\frac{k}{2^k} $$