The value of series $\sum_{n=1}^{\infty}\frac{n}{2^n}$ The value of series 
$\sum_{n=1}^{\infty}\frac{n}{2^n}$
I try to write some terms,but of no use. Is there any general method to approach such questions.
Thanks

Notice that
\begin{align*}
 \sum_{n=1}^\infty \frac{n}{2^n}
&= \frac{1}{2} \sum_{n=1}^\infty \frac{n}{2^{n-1}}
= \frac{1}{2} \sum_{n=0}^\infty \frac{n+1}{2^n}
= \frac{1}{2} \left( \sum_{n=0}^\infty \frac{n}{2^n}
   + \sum_{n=0}^\infty \frac{1}{2^n} \right) \\
&= \frac{1}{2} \sum_{n=1}^\infty \frac{n}{2^n} + 1,
\end{align*}
thus $\left( \sum_{n=1}^\infty \frac{n}{2^n} \right) = 2$.
