TITLE: Sum of the series $\sum_{1}^{\infty}\frac{n}{2^{n}}.$
QUESTION [0 upvotes]: Sum of the series $\sum_{1}^{\infty}\frac{n}{2^{n}}$ is 
$a.1$
$b.2$
$c.3$
$d.4$
With which known series I have to compare the given series? Please help. Thanks.

REPLY [4 votes]: Hint:
$${\frac {x}{(1-x)^{2}}}=\sum _{n=1}^{\infty }nx^{n}\quad {\text{ for }}|x|<1\!$$
let $x=\frac{1}{2}$ to get what you want
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