Find the Sum of an Infinite Series Compute
$${1 \cdot \frac {1}{2} + 2 \cdot \frac {1}{4} + 3 \cdot \frac {1}{8} + \dots + n \cdot \frac {1}{2^n} + \dotsb.}$$
I tried creating a partition for this but no such luck.
What would the the equation to generate the sum for the nth term

Let $S=\sum_{n\geq 1}\frac{n}{2^n}.$ Then 
$$2S=\sum_{n\geq 1}\frac{2n}{2^n}=\sum_{n\geq 1}\frac{n}{2^{n-1}}=\sum_{n\geq 1}\frac{(n-1)+1}{2^{n-1}} = 2+\sum_{n\geq 1}\frac{n-1}{2^{n-1}} = 2+S $$
hence $S=2$.
