Question

The value of ${\sum }_{n=0}^{\infty }n(\frac{1}{2}{)}^n$is

• A. 2

Answer 1

The correct option is A 2
$S={\sum }_{n=0}^{\infty }n(\frac{1}{2}{)}^n$
$S=1\left(\frac{1}{2}\right)+2(\frac{1}{2}{)}^2+3(\frac{1}{2}{)}^3+...$
$S=\frac{1}{2}[1+2(\frac{1}{2}+3(\frac{1}{2}{)}^2+4(\frac{1}{2}{)}^3+.......]$
$=\frac{1}{2}\left[\right(1-\frac{1}{2{)}^2}]$
$=\frac{1}{2}\left[\right(\frac{1}{2}{)}^{-2}]=\frac{2^2}{2}=2$