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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 Grandi's series ： ウィキペディア英語版 Grandi's series In mathematics, the infinite series $1\; -\; 1\; +\; 1\; -\; 1\; +\; \backslash dotsb$, also written :$$ \sum_^ (-1)^n is sometimes called Grandi's series, after Italian mathematician, philosopher, and priest Guido Grandi, who gave a memorable treatment of the series in 1703. It is a divergent series, meaning that it lacks a sum in the usual sense. On the other hand, its Cesàro sum is 1/2. ==Proof through integration== To prove using integration, first we consider the indefinite integral of $e^f(x)$ with respect to $x$ for any function $f$ by repeating integration by parts, with $\backslash frac=e^$ and $u=f(x),\; f\text{'}(x),\; f\text{'}\text{'}(x),\; \backslash dotsc$: $$ \begin \int e^x f(x)\, dx &= e^x f(x) - \int e^x f'(x)\, dx \\ &= e^x f(x) - e^x f'(x) + \int e^x f''(x)\, dx \\ &= e^x(f(x) - f'(x) + f''(x) - f(x) + \dotsb) + C \end Here, $C$ is the constant of integration. By letting $f(x)\; =\; \backslash sin\; x$ or $\backslash cos\; x$, we will get a result relevant to solving the Grandi series $G\; =\; 1\; -\; 1\; +\; 1\; -\; 1\; +\; \backslash dotsb$: $$ \begin \int e^x \sin x\, dx &= e^x(\sin x - \cos x - \sin x + \cos x + \sin x - \cos x - \sin x + \cos x + \dotsb) \\ &= e^x(\sin x (1 - 1 + 1 - 1 + \dotsb) + \cos x(-1 + 1 - 1 + 1 + \dotsb)) + C \\ &= Ge^x(\sin x - \cos x) + C\end We can also evaluate the integral using integration by parts with $\backslash frac=e^$ and $u\; =\; \backslash sin,\backslash cos$. We treat the integral as a function $I$ and realize that we have the same integral on both sides of the equation and add $I$ to both sides and divide by two. Symbolically: $$ \begin I &= \int e^x \sin x\, dx \\ &= e^x \sin x - \int e^ \cos x\, dx \\ &= e^x \sin x - e^x \cos x - \int e^x \sin x\, dx \\ &= e^x \sin x - e^x \cos x - I \\ &= \frac + C \end Comparing the results of evaluating the integral using both methods $\backslash int\; e^\backslash sin\; x\; \backslash ,\; dx=Ge^(\backslash sin-\backslash cos\; x)\; +\; C$ =\frac-\cos)} + C and thus $G=1\; -\; 1\; +\; 1\; -\; 1\; +\; \backslash dotsb\; =\; \backslash frac$. 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「Grandi's series」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.