Question

Evaluate the series

Answer 1

$S=\displaystyle\sum_{n=1}^7\left(\frac14\right)^n$

$\displaystyle S=\frac14+\left(\frac14\right)^2+\cdots+\left(\frac14\right)^7$
$\displaystyle\frac14S=\left(\frac14\right)^2+\left(\frac14\right)^3+\cdots+\left(\frac14\right)^8$
$\displaystyle S-\frac14S=\frac14+\left(\left(\frac14\right)^2-\left(\frac14\right)^2\right)+\cdots+\left(\left(\frac14\right)^7-\left(\frac14\right)^7\right)-\left(\frac14\right)^8$
$\displaystyle\frac34S=\frac14-\left(\frac14\right)^8$
$\displaystyle S=\frac{\frac14-\left(\frac14\right)^8}{\frac34}=\frac{5461}{16384}$