bookshelf/Bookshelf/Real/Sequence.tex

\documentclass{article}

\input{preamble}

\newcommand{\ns}{Real}
\newcommand{\linkA}[1]{\href{../Sequence/Arithmetic.html\#\ns.#1}{\ns.#1}}
\newcommand{\linkG}[1]{\href{../Sequence/Geometric.html\#\ns.#1}{\ns.#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{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{Geometric.sum\_recursive\_closed}

\end{proof}

\end{document}