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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 Ramanujan summation ： ウィキペディア英語版 Ramanujan summation Ramanujan summation is a technique invented by the mathematician Srinivasa Ramanujan for assigning a value to infinite divergent series. Although the Ramanujan summation of a divergent series is not a sum in the traditional sense, it has properties which make it mathematically useful in the study of divergent infinite series, for which conventional summation is undefined. == Summation == Ramanujan summation essentially is a property of the partial sums, rather than a property of the entire sum, as that doesn't exist. If we take the Euler–Maclaurin summation formula together with the correction rule using Bernoulli numbers, we see that: :$\backslash begin$ f\left( 0\right) + f\left( 1\right) + \cdots + f\left( n - 1\right) + \fracf\left( n\right) \\ = &\frac\left(0\right) + f\left( n\right)\right ) + \sum_^ f \left( k \right) \\ = &\int_0^n f(x)\,dx + \sum_^p \frac\left(- f^(0)\right ) + R_p \end Ramanujan〔 Bruce C. Berndt, (Ramanujan's Notebooks ), ''Ramanujan's Theory of Divergent Series'', Chapter 6, Springer-Verlag (ed.), (1939), pp. 133-149.〕 wrote it for the case ''p'' going to infinity: :$\backslash sum\_^f(k)\; =\; C\; +\; \backslash int\_0^x\; f(t)\backslash ,dt\; +\; \backslash fracf(x)\; +\; \backslash sum\_^\backslash fracf^(x)$ where ''C'' is a constant specific to the series and its analytic continuation and the limits on the integral were not specified by Ramanujan, but presumably they were as given above. Comparing both formulae and assuming that ''R'' tends to 0 as ''x'' tends to infinity, we see that, in a general case, for functions ''f''(''x'') with no divergence at ''x'' = 0: :$C(a)=\backslash int\_0^a\; f(t)\backslash ,dt-\backslash fracf(0)-\backslash sum\_^\backslash fracf^(0)$ where Ramanujan assumed $\backslash scriptstyle\; a\; \backslash ,=\backslash ,\; 0$. By taking $\backslash scriptstyle\; a\; \backslash ,=\backslash ,\; \backslash infty$ we normally recover the usual summation for convergent series. For functions ''f''(''x'') with no divergence at ''x'' = 1, we obtain: :$C(a)\; =\; \backslash int\_1^a\; f(t)\backslash ,dt+\; \backslash fracf(1)\; -\; \backslash sum\_^\backslash fracf^(1)$ ''C''(0) was then proposed to use as the sum of the divergent sequence. It is like a bridge between summation and integration. The convergent version of summation for functions with appropriate growth condition is then: :$f(1)+f(2)+f(3)+...=-\backslash frac+i\backslash int\_^\backslash infty\; \backslash frac\; dt$ To compare see Abel-Plana 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「Ramanujan summation」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.