Question

Find the sum of the series. 1 + z/5 + z^2/25 + z^3.125

Answer 1

`1+\frac z5+(z^2)/(25)+(z^3)/(125)+\cdots=\displaystyle\sum_(k=0)^\infty\left(\frac z5\right)^k`

`=\frac1{1-\frac z5}=\frac5{5-z}`

provided that
`\left|\frac z5\right|<1\iff|z|<5`.