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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 Liouville number ： ウィキペディア英語版 Liouville number In number theory, a Liouville number is an irrational number ''x'' with the property that, for every positive integer ''n'', there exist integers ''p'' and ''q'' with ''q'' > 1 and such that : A Liouville number can thus be approximated "quite closely" by a sequence of rational numbers. In 1844, Joseph Liouville showed that all Liouville numbers are transcendental, thus establishing the existence of transcendental numbers for the first time. == The existence of Liouville numbers (Liouville's constant) == Here we show that Liouville numbers exist by exhibiting a construction that produces such numbers. For any integer ''b'' ≥ 2, and any sequence of integers (''a''1, ''a''2, …, ), such that ''a''''k'' ∈ , ∀''k'' ∈ , define the number :$x\; =\; \backslash sum\_^\backslash infty\; \backslash frac\backslash ,;\; \backslash quad\; p\_n\; =\; q\_n\; \backslash sum\_^n\; \backslash frac\backslash right|\; =\; \backslash sum\_^\backslash infty\; \backslash frac^\backslash infty\; \backslash frac^\backslash infty\; \backslash frac\; =\; \backslash frac^\backslash infty\; \backslash frac\; =\; \backslash frac\; =\; \backslash frac\}^n\}\backslash ,,$ ...where the last equality follows from the fact that :$n\backslash cdot\; n!\; =\; n\backslash cdot\; n!\; +\; n!\; -\; n!\; =\; (n+1)!\; -\; n!\backslash ;.$ Therefore, we conclude that any such ''x'' is a Liouville number. 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「Liouville number」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.