Question

What is the sum to infinity of the following series ?
pic shows quesion

Answer 1

`\frac14+\frac2{4^2}+\frac1{4^3}+\frac2{4^4}+\cdots`

`=\left(\frac14+\frac1{4^2}+\frac1{4^3}+\frac1{4^4}+\cdots\right)+\left(\frac1{4^2}+\frac1{4^4}+\cdots\right)`

`=\displaystyle\sum_(n=1)^\infty\left(\frac14\right)^n+\sum_(n=1)^\infty\left(\frac14\right)^(2n)`

`=\displaystyle\sum_(n=0)^\infty\left(\frac14\right)^n-1+\sum_(n=0)^\infty\left(\frac14\right)^(2n)-1`

`=\displaystyle\sum_(n=0)^\infty\left(\frac14\right)^n+\sum_(n=0)^\infty\left(\frac1{16}\right)^n-2`

`=\frac1{1-\frac14}+\frac1{1-\frac1{16}}-2`

`=\frac25`