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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 Infinite product ： ウィキペディア英語版 Infinite product In mathematics, for a sequence of complex numbers ''a''1, ''a''2, ''a''3, ... the infinite product :$$ \prod_^ a_n = a_1 \; a_2 \; a_3 \cdots is defined to be the limit of the partial products ''a''1''a''2...''a''''n'' as ''n'' increases without bound. The product is said to ''converge'' when the limit exists and is not zero. Otherwise the product is said to ''diverge''. A limit of zero is treated specially in order to obtain results analogous to those for infinite sums. Some sources allow convergence to 0 if there are only a finite number of zero factors and the product of the non-zero factors is non-zero, but for simplicity we will not allow that here. If the product converges, then the limit of the sequence ''a''''n'' as ''n'' increases without bound must be 1, while the converse is in general not true. The best known examples of infinite products are probably some of the formulae for π, such as the following two products, respectively by Viète (Viète's formula, the first published infinite product in mathematics) and John Wallis (Wallis product): :$\backslash frac\; =\; \backslash frac\; \backslash cdot\; \backslash frac\; \backslash cdot\; \backslash frac\; \}\; \backslash cdots$ :$\backslash frac\; =\; \backslash frac\; \backslash cdot\; \backslash frac\; \backslash cdot\; \backslash frac\; \backslash cdot\; \backslash frac\; \backslash cdot\; \backslash frac\; \backslash cdot\; \backslash frac\; \backslash cdot\; \backslash frac\; \backslash cdot\; \backslash frac\; \backslash cdots\; =\; \backslash prod\_^\; \backslash left(\; \backslash frac\; \backslash right).$ == Convergence criteria == The product of positive real numbers :$\backslash prod\_^\; a\_n$ converges to a nonzero real number if and only if the sum :$\backslash sum\_^\; \backslash ln(a\_n)$ converges. This allows the translation of convergence criteria for infinite sums into convergence criteria for infinite products. The same criterion applies to products of arbitrary complex numbers (including negative reals) if the logarithm is understood as a fixed branch of logarithm which satisfies ln(1) = 0, with the proviso that the infinite product diverges when infinitely many ''an'' fall outside the domain of ln, whereas finitely many such ''an'' can be ignored in the sum. For products of reals in which each $a\_n\backslash ge1$, written as, for instance, $a\_n=1+p\_n$, where $p\_n\backslash ge\; 0$, the bounds :$1+\backslash sum\_^\; p\_n\; \backslash le\; \backslash prod\_^\; \backslash left(\; 1\; +\; p\_n\; \backslash right)\; \backslash le\; \backslash exp\; \backslash left(\; \backslash sum\_^p\_n\; \backslash right)$ show that the infinite product converges precisely if the infinite sum of the ''p''''n'' converges. This relies on the Monotone convergence theorem. More generally, the convergence of $\backslash prod\_^\backslash infty(1+p\_n)$ is equivalent to the convergence of $\backslash sum\_^\backslash infty\; p\_n$ if ''pn'' are real or complex numbers such that , since $\backslash ln(1+x)=x+O(x^2)$ in a neighbourhood of 0. If the series ''p''''n'' diverges to zero, then the sequence of partial products of the ''p''''n'' converges to zero as a sequence. The infinite product is said to diverge to zero. 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「Infinite product」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.