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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 Reciprocal Fibonacci constant ： ウィキペディア英語版 Reciprocal Fibonacci constant The reciprocal Fibonacci constant, or ψ, is defined as the sum of the reciprocals of the Fibonacci numbers: :$\backslash psi\; =\; \backslash sum\_^\; \backslash frac\; =\; \backslash frac\; +\; \backslash frac\; +\; \backslash frac\; +\; \backslash frac\; +\; \backslash frac\; +\; \backslash frac\; +\; \backslash frac\; +\; \backslash frac\; +\; \backslash cdots.$ The ratio of successive terms in this sum tends to the reciprocal of the golden ratio. Since this is less than 1, the ratio test shows that the sum converges. The value of ψ is known to be approximately :$\backslash psi\; =\; 3.359885666243177553172011302918927179688905133732\backslash dots\; .$ Gosper describes an algorithm for fast numerical approximation of its value. The reciprocal Fibonacci series itself provides O(''k'') digits of accuracy for ''k'' terms of expansion, while Gosper's accelerated series provides O(''k''2) digits.〔.〕 ψ is known to be irrational; this property was conjectured by Paul Erdős, Ronald Graham, and Leonard Carlitz, and proved in 1989 by Richard André-Jeannin. The continued fraction representation of the constant is: : $\backslash psi\; =(\backslash dots)\; \backslash !\backslash ,\; .$ ==See also== *List of sums of reciprocals 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「Reciprocal Fibonacci constant」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.