\begin{definition}[Definition:Geometric Series]
Let $\sequence {x_n}$ be a geometric sequence in $\R$:
:$x_n = a r^n$ for $n = 0, 1, 2, \ldots$
Then the series defined as:
:$\ds \sum_{n \mathop = 0}^\infty x_n = a + a r + a r^2 + \cdots + a r^n + \cdots$
is a '''geometric series'''.
\end{definition}