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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 1 − 1 + 2 − 6 + 24 − 120 + ... ： ウィキペディア英語版 1 − 1 + 2 − 6 + 24 − 120 + ... In mathematics, the divergent series :$\backslash sum\_^\backslash infty\; (-1)^k\; k!$ was first considered by Euler, who applied summability methods to assign a finite value to the series.〔〕 The series is a sum of factorials that alternatingly are added or subtracted. A way to assign a value to the divergent series is by using Borel summation, where we formally write :$\backslash sum\_^\backslash infty\; (-1)^k\; k!\; =\; \backslash sum\_^\backslash infty\; (-1)^k\; \backslash int\_0^\backslash infty\; x^k\; \backslash exp(-x)\; \backslash ,\; dx$ If we interchange summation and integration (ignoring the fact that neither side converges), we obtain: :$\backslash sum\_^\backslash infty\; (-1)^k\; k!\; =\; \backslash int\_0^\backslash infty\; \backslash left((-x)^k\; \backslash right)\backslash exp(-x)\; \backslash ,\; dx$ The summation in the square brackets converges and equals 1/(1 + ''x'') if ''x'' < 1. If we analytically continue this 1/(1 + ''x'') for all real ''x'', we obtain a convergent integral for the summation: :$\backslash sum\_^\backslash infty\; (-1)^\; k!\; =\; \backslash int\_0^\backslash infty\; \backslash frac\; \backslash ,\; dx\; =\; e\; E\_1\; (1)\; \backslash approx\; 0.596347362323194074341078499369\backslash ldots$ where $E\_1\; (z)$ is the exponential integral. This is by definition the Borel sum of the series. ==Derivation== Consider the coupled system of differential equations :$\backslash dot(t)\; =\; x(t)\; -\; y(t),\backslash qquad\; \backslash dot(t)=-y(t)^$ where dots denote time derivatives. The solution with stable equilibrium at $(x,y)=(0,0)$ as $t\backslash to\backslash infty$ has $y(t)=1/t$. And substituting it into the first equation gives us a formal series solution :$x(t)\; =\; \backslash sum^\_(-1)^\backslash frac\backslash int^\_\backslash frac\backslash mathrmu.$ By successively integrating by parts, we recover the formal power series as an asymptotic approximation to this expression for $x(t)$. Euler argues (more or less) that setting equals to equals gives us :$\backslash sum^\_(-1)^(n-1)!\; =\; e\backslash int^\_\backslash frac\backslash mathrmu.$ 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「1 − 1 + 2 − 6 + 24 − 120 + ...」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.