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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 Stolz–Cesàro theorem ： ウィキペディア英語版 Stolz–Cesàro theorem In mathematics, the Stolz–Cesàro theorem, named after mathematicians Otto Stolz and Ernesto Cesàro, is a criterion for proving the convergence of a sequence. Let $(a\_n)\_$ and $(b\_n)\_$ be two sequences of real numbers. Assume that $(b\_n)\_$ is strictly monotone and divergent sequence (i.e. strictly increasing and approaches $+\; \backslash infty$ or strictly decreasing and approaches $-\; \backslash infty$) and the following limit exists: :$\backslash lim\_\; \backslash frac=\backslash ell.\backslash$ Then, the limit :$\backslash lim\_\; \backslash frac\backslash$ also exists and it is equal to ''ℓ''. The general form of the Stolz–Cesàro theorem is the following (see http://www.imomath.com/index.php?options=686): If $(a\_n)\_$ and $(b\_n)\_$ are two sequences such that $(b\_n)\_$ is monotone and unbounded, then: :$\backslash liminf\_\; \backslash frac\backslash leq\; \backslash liminf\_\backslash frac\backslash leq\backslash limsup\_\backslash frac\backslash leq\backslash limsup\_\backslash frac.$ The Stolz–Cesàro theorem can be viewed as a generalization of the Cesàro mean, but also as a l'Hôpital's rule for sequences. The ∞/∞ case is stated and proved on pages 173—175 of Stolz's 1885 book S and also on page 54 of Cesàro's 1888 article C. It appears as Problem 70 in Pólya and Szegő. ==References== *. *. *. *. 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「Stolz–Cesàro theorem」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.