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\section*{Sum of Geometric Series}%
\label{sec:sum-geometric-series}

Let $(a_i)_{i \geq 0}$ be a geometric sequence with common ratio $r \neq 1$.
Then for some $n \in \mathbb{N}$,
$$\sum_{i=0}^n a_i = \frac{a_0(1 - r^{n+1})}{1 - r}.$$

\begin{proof}

  \link{sum_recursive_closed}

\end{proof}

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