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}
$