How to find the sum of this series?

The series is $1\cdot\frac{1}{2} + 2\cdot\frac{1}{4} + 3\cdot\frac{1}{8} + \cdots$

Or in other words

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

What kind of series is this and how to find the sum? Thanks....

Answer

Without calculus:

If
$s(a) =\sum_{n=0}^{\infty} na^n$
for
$|a| < 1$,
then

$\begin{array}\\ as(a) &=\sum_{n=0}^{\infty} na^{n+1}\\ &=\sum_{n=1}^{\infty} (n-1)a^{n}\\ &=\sum_{n=1}^{\infty} na^{n}-\sum_{n=1}^{\infty} a^{n}\\ &=s(a)-\dfrac{a}{1-a}\\ \text{so}\\ s(a) &=\dfrac{a}{(1-a)^2}\\ \end{array}$