A geometric series is typically written as
$a + ar + ar^2 + \dots$
Using summation notation, the geometric series can be expressed as
$\sum_{k=1}^\infty ar^{k-1}$
We know from [[calculus]] that the sum can be expressed as
$\sum_{k=1}^\infty ar^{k-1} =
\begin{cases}
\frac{a}{1-r} & |r| < 1 \\
\text diverges & |r| >= 1
\end{cases}
$