bookshelf/Bookshelf/Real/Sequence.tex

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\documentclass{article}
\input{preamble}
\newcommand{\linkA}[1]{\href{../../../Bookshelf/Real/Sequence/Arithmetic.html\##1}{#1}}
\newcommand{\linkG}[1]{\href{../../../Bookshelf/Real/Sequence/Geometric.html\##1}{#1}}
\begin{document}
\section*{Sum of Arithmetic Series}%
\label{sec:sum-arithmetic-series}
Let $(a_i)_{i \geq 0}$ be an arithmetic sequence with common difference $d$.
Then for some $n \in \mathbb{N}$,
$$\sum_{i=0}^n a_i = \frac{(n + 1)(a_0 + a_n)}{2}.$$
\begin{proof}
\linkA{Real.Arithmetic.sum\_recursive\_closed}
\end{proof}
\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}
\linkG{Real.Geometric.sum\_recursive\_closed}
\end{proof}
\end{document}