TITLE: How do you prove $\sum \frac {n}{2^n} = 2$?
QUESTION [6 upvotes]: How do you prove $$\sum_{n=1}^{\infty} \frac {n}{2^n} = 2\ ?$$
My attempt: I have been trying to find geometric series that converge to 2 which can bind the given series on either side. But I am unable to find these. Is there a general technique to find the sum? This is a high school interview question and must be easy enough to solve in a few minutes.
Please give any hints for the first step towards a solution.

REPLY [3 votes]: A general way to handle similar problems is to note that 
$$\sum_{n=1}^{\infty}nx^n=x\frac{d}{dx}\sum_{n=1}^{\infty}x^n \tag1$$
The sum on the right-hand side of $(1)$ is the series for
$$\sum_{n=1}^{\infty}x^n=\frac{x}{1-x}\tag 2$$
Taking a derivative of the right-hand side of $(2)$ reveals that 
$$\sum_{n=1}^{\infty}nx^n=\frac{x}{(1-x)^2}$$
whereupon evaluation at $x=1/2$ provide the expected result 

$$\sum_{n=1}^{\infty}\frac{n}{2^n}=2$$