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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 Euler's continued fraction formula ： ウィキペディア英語版 Euler's continued fraction formula In the analytic theory of continued fractions, Euler's continued fraction formula is an identity connecting a certain very general infinite series with an infinite continued fraction. First published in 1748, it was at first regarded as a simple identity connecting a finite sum with a finite continued fraction in such a way that the extension to the infinite case was immediately apparent.〔1748 Leonhard Euler, ''Introductio in analysin infinitorum'', Vol. I, Chapter 18.〕 Today it is more fully appreciated as a useful tool in analytic attacks on the general convergence problem for infinite continued fractions with complex elements. == The original formula == Euler derived the formula as an identity connecting a finite sum of products with a finite continued fraction. :$$ a_0 + a_0a_1 + a_0a_1a_2 + \cdots + a_0a_1a_2\cdots a_n = \cfrac}}}}}}\, The identity is easily established by induction on ''n'', and is therefore applicable in the limit: if the expression on the left is extended to represent a convergent infinite series, the expression on the right can also be extended to represent a convergent infinite continued fraction. 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「Euler's continued fraction formula」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.