Question

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

Answer 1

$1+\dfrac z5+\dfrac{z^2}{25}+\dfrac{z^3}{125}+\cdots=\displaystyle\sum_{k=0}^\infty\left(\frac z5\right)^k$
$=\dfrac1{1-\frac z5}=\dfrac5{5-z}$

provided that $\left|\dfrac z5\right|<1\iff|z|<5$.