Answer
Please see step-by-step
Work Step by Step
The series
$\displaystyle \sum_{n=0}^{\infty}ar^{n}=a+ar+ar^{2}+\cdots+ar^{n}+\cdots,\ a\neq 0$
is a geometric series with ratio $r$.
By Th.9.6,
When $0 \lt |r| \lt 1$, the series converges to $\displaystyle \frac{a}{1-r}$.
When $|r|\geq 1$, the series diverges.