bookshelf/common/Common/Data/Real/Sequence.tex

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\documentclass{article}
\input{../../../preamble}
\begin{document}
\begin{theorem}[Sum of Arithmetic Series]
Let $(a_i)_{i \geq 0}$ be an arithmetic sequence with common difference $d$.
Then for some $n \in \mathbb{N}$,
$$\sum_{i=0}^n a_i = \frac{(n + 1)(a_0 + a_n)}{2}.$$
\end{theorem}
\begin{proof}
\href{Sequence.lean}{Common.Data.Real.Sequence.Arithmetic.sum_recursive_closed}
\end{proof}
\begin{theorem}[Sum of Geometric Series]
Let $(a_i)_{i \geq 0}$ be a geometric sequence with common ratio $r \neq 1$.
Then for some $n \in \mathbb{N}$,
$$\sum_{i=0}^n a_i = \frac{a_0(1 - r^{n+1})}{1 - r}.$$
\end{theorem}
\begin{proof}
\href{Sequence.lean}{Common.Data.Real.Sequence.Geometric.sum_recursive_closed}
\end{proof}
\end{document}