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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 Wallis' integrals ： ウィキペディア英語版 Wallis' integrals In mathematics, and more precisely in analysis, the Wallis' integrals constitute a family of integrals introduced by John Wallis. == Definition, basic properties == The ''Wallis' integrals'' are the terms of the sequence $(W\_n)\_$ defined by: :$W\_n\; =\; \backslash int\_0^\}\; \backslash sin^n(x)\backslash ,dx,$ or equivalently (through a substitution: $x\; =\; \backslash frac\; -\; t$): :$W\_n\; =\; \backslash int\_0^\}\; \backslash cos^n(x)\backslash ,dx$ In particular, the first few terms of this sequence are: | align="center" | $1$ | align="center" | $\backslash frac$ | align="center" | $\backslash frac$ | align="center" | $\backslash frac$ | align="center" | $\backslash frac$ | align="center" | $\backslash frac$ | align="center" | $\backslash frac$ | align="center" | $\backslash frac$ | align="center" | ... |} The sequence $\backslash \; (W\_n)$ is decreasing and has strictly positive terms. In fact, for all $n\; \backslash in\backslash ,\; \backslash mathbb$ : *$\backslash \; W\_n\; >\; 0$, because it is an integral of a non-negative continuous function which is not all zero in the integration interval *$W\_\; -\; W\_=\; \backslash int\_0^\}\; \backslash sin^(x)\backslash ,dx\; -\; \backslash int\_0^\}\; \backslash sin^(x)\backslash ,dx\; =\; \backslash int\_0^\}\; \backslash sin^(x)\backslash ,\; (-\; \backslash sin(x))\backslash ,dx\; \backslash geqslant\; 0$ :(by the linearity of integration and because the last integral is an integral of a non-negative function within the integration interval) Note: Since the sequence $\backslash \; (W\_n)$ is decreasing and bounded below by 0, it converges to a non-negative limit. Indeed, the limit is zero (see below). 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「Wallis' integrals」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.