Question

Value of
${\sum }_{k=1}^{\infty }{\sum }_{r=0}^k\frac{1}{3^k}\left({{}^{k}C}_r\right)$ is

• A. $2$
• B. $\frac{2}{3}$
• C. $\frac{1}{3}$
• D. none of these

Answer 1

The correct option is A $2$

$\sum \frac{1}{3^k}{}^k{C}_r$
$=\frac{1}{3^k}\sum {}^k{C}_r$
$=\frac{2^k}{3^k}$
This is a G.P
Therefore, the sum of the series will be
$S=\frac{\frac{2}{3}}{1-\frac{2}{3}}$ $=2$