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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 Sophomore's dream ： ウィキペディア英語版 Sophomore's dream In mathematics, sophomore's dream is a name occasionally used for the identities (especially the first) :$\backslash begin$ \int_0^1 x^\,\mathrmx &= \sum_^\infty n^&&(\scriptstyle \\ \int_0^1 x^x \,\mathrmx &= \sum_^\infty (-1)^n^ = - \sum_^\infty (-n)^ &&(\scriptstyle) \end discovered in 1697 by Johann Bernoulli. The name "sophomore's dream", which appears in , is in contrast to the name "freshman's dream" which is given to the incorrect〔Incorrect unless one is working over a field or unital commutative ring of prime characteristic ''n'' or a factor of ''n''. The correct result is given by the binomial theorem.〕 equation . The sophomore's dream has a similar too-good-to-be-true feel, but is in fact true. == Proof == We prove the second identity; the first is completely analogous. The key ingredients of the proof are: * Write ''x''''x'' = exp(''x'' log ''x''). * Expand exp(''x'' log ''x'') using the power series for exp. * Integrate termwise. * Integrate by substitution. Expand ''x''''x'' as : $x^x\; =\; \backslash exp(x\; \backslash log\; x)\; =\; \backslash sum\_^\backslash infty\; \backslash frac.$ Therefore, we have : $\backslash int\_0^1\; x^x\backslash ,\backslash mathrmx\; =\; \backslash int\_0^1\; \backslash sum\_^\backslash infty\; \backslash frac\; \backslash ,\backslash mathrmx.$ By uniform convergence of the power series, we may interchange summation and integration : $\backslash int\_0^1\; x^x\backslash ,\backslash mathrmx\; =\; \backslash sum\_^\backslash infty\; \backslash int\_0^1\; \backslash frac\; \backslash ,\backslash mathrmx.$ To evaluate the above integrals we perform the change of variable in the integral $\backslash scriptstyle\; x=\backslash exp\backslash ,\; \backslash left(-\backslash frac\backslash right)$, with , giving us :$\backslash int\_0^1\; x^n(\backslash log\backslash ,\; x)^n\backslash ,\backslash mathrmx\; =\; (-1)^n\; (n+1)^\; \backslash int\_0^\backslash infty\; u^n\; e^\backslash ,\backslash mathrmu.$ By the well-known Euler's integral identity for the Gamma function :$\backslash int\_0^\backslash infty\; u^n\; e^\backslash ,\backslash mathrmu=n!$ so that: :$\backslash int\_0^1\; \backslash frac\backslash ,\backslash mathrmx$ = (-1)^n (n+1)^. Summing these (and changing indexing so it starts at ''n'' = 1 instead of ''n'' = 0) yields the formula. 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「Sophomore's dream」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.