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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 Riemann series theorem ： ウィキペディア英語版 Riemann series theorem In mathematics, the Riemann series theorem (also called the Riemann rearrangement theorem), named after 19th-century German mathematician Bernhard Riemann, says that if an infinite series is conditionally convergent, then its terms can be arranged in a permutation so that the new series converges to any given value, or diverges. ==Definitions== A series $\backslash sum\_^\backslash infty\; a\_n$ converges if there exists a value $\backslash ell$ such that the sequence of the partial sums :$\backslash left\; \backslash ,\; \backslash quad\; S\_n\; =\; \backslash sum\_^n\; a\_k,$ converges to $\backslash ell$. That is, for any ''ε'' > 0, there exists an integer ''N'' such that if ''n'' ≥ ''N'', then :$\backslash left\backslash vert\; S\_n\; -\; \backslash ell\; \backslash right\backslash vert\; \backslash le\; \backslash \; \backslash epsilon.$ A series converges conditionally if the series $\backslash sum\_^\backslash infty\; a\_n$ converges but the series $\backslash sum\_^\backslash infty\; \backslash left\backslash vert\; a\_n\; \backslash right\backslash vert$ diverges. A permutation is simply a bijection from the set of positive integers to itself. This means that if $\backslash sigma$ is a permutation, then for any positive integer $b$, there exists exactly one positive integer $a$ such that $\backslash sigma\; (a)\; =\; b$. In particular, if $x\; \backslash ne\; y$, then $\backslash sigma\; (x)\; \backslash ne\; \backslash sigma\; (y)$. 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「Riemann series 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.