Simplification of geometric series. Can someone please help simplify this series?
$$\sum_{k=1}^\infty k\left(\frac 12\right)^k$$
In general, $$\sum_{k=1}^\infty\left(\frac 12\right)^k = \frac{1}{1-\frac{1}{2}} =2.$$
However, I am confused with the $k$ in front of the term $k\big(\frac 12\big)^k$.
I understand if the problem is $\sum_{k=1}^\infty\big(\frac 12\big)^k$. However, it is the $k$ term that I don't understand. The answer is suppose to be $2$.
Can someone help me? Thank you.

We have the geometric series
$$
\sum_{k=0}^\infty x^k=\frac1{1-x}
$$
Differentiate to get
$$
\sum_{k=0}^\infty kx^{k-1}=\frac1{(1-x)^2}
$$
Multiply by $x$, and plug in $x=\frac12$.
