Chapter 6.2, Problem 93E

In the following exercises, evaluate each infinite series by identifying it as the value of a derivative or integral of geometric series.

93. Evaluate ${\displaystyle \underset{n=2}{\overset{\infty }{\sum }}\frac{n(n-1)}{2^n}}$ as $f\text{'}\left(\frac{1}{2}\right)$ where $f(x)={\displaystyle \underset{n=0}{\overset{\infty }{\sum }}{x}^n.}$