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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 Faulhaber's formula ： ウィキペディア英語版 Faulhaber's formula In mathematics, Faulhaber's formula, named after Johann Faulhaber, expresses the sum of the ''p''-th powers of the first ''n'' positive integers :$\backslash sum\_^n\; k^p\; =\; 1^p\; +\; 2^p\; +\; 3^p\; +\; \backslash cdots\; +\; n^p$ as a (''p'' + 1)th-degree polynomial function of ''n'', the coefficients involving Bernoulli numbers ''Bj''. The formula says :$\backslash sum\_^n\; k^p\; =\; \backslash sum\_^p\; (-1)^j\; B\_j\; n^,\backslash qquad\; \backslash mbox~B\_1\; =\; -\backslash frac.$ Faulhaber himself did not know the formula in this form, but only computed the first seventeen polynomials; the general form was established with the discovery of the Bernoulli numbers (see History section below). The derivation of Faulhaber's formula is available in ''The Book of Numbers'' by John Horton Conway and Richard K. Guy. There is also a similar (but somehow simpler) expression: using the idea of telescoping and the binomial theorem, one gets ''Pascal's identity'': :$(n+1)^\; -\; 1\; =\; \backslash sum\_^n\; \backslash left((m+1)^\; -\; m^\backslash right)\; =\; \backslash sum\_^k\; \backslash binom\; (1^p+2^p+\; \backslash dots\; +\; n^p)$. This in particular yields the examples below, e.g., take ''k'' = 1 to get the first example. ==Examples== :$1\; +\; 2\; +\; 3\; +\; \backslash cdots\; +\; n\; =\; \backslash frac\; =\; \backslash frac$ (the triangular numbers) :$1^2\; +\; 2^2\; +\; 3^2\; +\; \backslash cdots\; +\; n^2\; =\; \backslash frac\; =\; \backslash frac$ (the square pyramidal numbers) :$1^3\; +\; 2^3\; +\; 3^3\; +\; \backslash cdots\; +\; n^3\; =\; \backslash left()^2\; =\; \backslash frac$ (the squared triangular numbers) :$$ \begin 1^4 + 2^4 + 3^4 + \cdots + n^4 & = \frac \\ & = \frac \end :$$ \begin 1^5 + 2^5 + 3^5 + \cdots + n^5 & = \frac \\ & = \frac \end :$$ \begin 1^6 + 2^6 + 3^6 + \cdots + n^6 & = \frac \\ & = \frac \end 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「Faulhaber's formula」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.