45 lines
950 B
TeX
45 lines
950 B
TeX
\documentclass{article}
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\input{../../preamble}
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\newcommand{\lean}[2]{\leanref{../../#1.html\##2}{#2}}
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\begin{document}
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\header{Sequences}{}
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\tableofcontents
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\section{Summations}%
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\label{sec:summations}
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\subsection{\unverified{Arithmetic Series}}%
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\label{sub: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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\lean{Common/Real/Sequence/Arithmetic}
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{Real.Arithmetic.sum\_recursive\_closed}
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\end{proof}
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\subsection{\unverified{Geometric Series}}%
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\label{sub: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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\lean{Common/Real/Sequence/Geometric}
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{Real.Geometric.sum\_recursive\_closed}
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\end{proof}
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\end{document}
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