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=(\frac14-\left(\frac14\right)^8)/(\frac34)=(5461)/(16384)`