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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 D'Alembert criterion ： ウィキペディア英語版 Ratio test In mathematics, the ratio test is a test (or "criterion") for the convergence of a series :$\backslash sum\_^\backslash infty\; a\_n,$ where each term is a real or complex number and is nonzero when is large. The test was first published by Jean le Rond d'Alembert and is sometimes known as d'Alembert's ratio test or as the Cauchy ratio test.〔http://mathworld.wolfram.com/RatioTest.html〕 == Motivation == Given the following geometric series: :$\backslash sum\_^\backslash infty\; \backslash left(\backslash frac\backslash right)^n\; =\; \backslash frac\; +\; \backslash frac\; +\; \backslash frac\; +\; \backslash cdots$ The quotient :$a\_/a\_n\; =\; (1/2)^/(1/2)^n$ of any two adjacent terms is 1/2. The sum of the first ''m'' terms is given by: :$1\; -\; \backslash frac.$ As ''m'' increases, this converges to 1, so the sum of the series is 1. On the other hand given this geometric series: :$\backslash sum\_^\backslash infty\; 2^n\; =\; 2\; +\; 4\; +\; 8\; +\; \backslash cdots$ The quotient $a\_/a\_n$ of any two adjacent terms is 2. The sum of the first ''m'' terms is given by :$2^\; -\; 2,$ which increases without bound as ''m'' increases, so this series diverges. More generally, the sum of the first ''m'' terms of the geometric series is given by: :$\backslash sum\_^\; r^n\; =\; \backslash frac\; (r^m\; -\; 1).$ Whether this converges or diverges as ''m'' increases depends on whether ''r'', the quotient of any two adjacent terms, is less than or greater than 1. Now consider the series: :$\backslash sum\_^\backslash infty\; \backslash frac\; \backslash left(\backslash frac\backslash right)^n\; =\; \backslash frac\; \backslash cdot\; \backslash frac\; +\; \backslash frac\; \backslash cdot\; \backslash frac\; +\; \backslash frac\; \backslash cdot\; \backslash frac\; +\; \backslash cdots$ This is similar to the first convergent sequence above, except that now the ratio of two terms is not fixed at exactly 1/2: :$\backslash left(\backslash frac\; \backslash left(\backslash frac\backslash right)^n\backslash right)/\backslash left(\backslash frac\; \backslash left(\backslash frac\backslash right)^\backslash right)\; =\; \backslash frac\; =\; \backslash frac\; -\; \backslash frac.$ However, as ''n'' increases, the ratio still tends in the limit towards the same constant 1/2. The ratio test generalizes the simple test for geometric series to more complex series like this one where the quotient of two terms is not fixed, but in the limit tends towards a fixed value. The rules are similar: if the quotient approaches a value less than one, the series converges, whereas if it approaches a value greater than one, the series diverges. 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「Ratio 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.