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