Answer
See below.
Work Step by Step
A geometric series is a series of the form
$ a+ar+ar^{2}+\displaystyle \cdots+ar^{n-1}+\cdots=\sum_{n=1}^{\infty}ar^{n-1},\quad a\neq 0$.
The parameter $r$ is called the common ratio of the series and $a$ is the first term of the sum of the series.
It converges to
$\displaystyle \frac{a}{1-r}$ if $|r|\lt 1$
and diverges if
$|r|\geq 1.$
Example:
The series
$\displaystyle \sum_{n=1}^{\infty}2\cdot 3^{n-1}$
diverges, because
$|r|=|3|\geq 1$
Example
$\displaystyle \sum_{n=1}^{\infty}2\cdot(0.5)^{n-1}$
converges because
$|r|=|0.5|\lt 1$
and, the sum equals
$\displaystyle \frac{2}{1-0.5}=\frac{2}{0.5}=4$
