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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Alternating series test ： ウィキペディア英語版 Alternating series test In mathematical analysis, the alternating series test is a method used to prove that an alternating series with terms that decrease in absolute value is a convergent series. The test was used by Gottfried Leibniz and is sometimes known as Leibniz's test, Leibniz's rule, or the Leibniz criterion. == Formulation == A series of the form :$\backslash sum\_^\backslash infty\; (-1)^\; a\_n\; =\; a\_1\; -\; a\_2\; +\; a\_3\; -\; \backslash cdots\; \backslash !$ Or, :$\backslash sum\_^\backslash infty\; (-1)^\; a\_n\; =\; -\; a\_1\; +\; a\_2\; -\; a\_3\; +\; \backslash cdots\; \backslash !$ where ''a''''n'' are positive, is called an alternating series. The alternating series test then says: if $a\_n$ decreases monotonically and $\backslash lim\_\; a\_n\; =\; 0$ then the alternating series converges. Moreover, let ''L'' denote the sum of the series, then the partial sum :$S\_k\; =\; \backslash sum\_^k\; (-1)^\; a\_n\backslash !$ approximates ''L'' with error bounded by the next omitted term: :$\backslash left\; |\; S\_k\; -\; L\; \backslash right\; \backslash vert\; \backslash le\; \backslash left\; |\; S\_k\; -\; S\_\; \backslash right\; \backslash vert\; =\; a\_.\backslash !$ ==Proof== Suppose we are given a series of the form $\backslash sum\_^\backslash infty\; (-1)^\; a\_n\backslash !$, where $\backslash lim\_a\_=0$ and $a\_n\; \backslash geq\; a\_$ for all natural numbers ''n''. (The case $\backslash sum\_^\backslash infty\; (-1)^\; a\_n\backslash !$ follows by taking the negative.) 〔 The proof follows the idea given by James Stewart (2012) “Calculus: Early Transcendentals, Seventh Edition” pp. 727–730. ISBN 0-538-49790-4〕 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Alternating series test」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.