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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 expression (mathematics) ： ウィキペディア英語版 Infinite expression (mathematics) In mathematics, an infinite expression is an expression in which some operators take an infinite number of arguments, or in which the nesting of the operators continues to an infinite depth.〔.〕 A generic concept for infinite expression can lead to ill-defined or self-inconsistent constructions (much like a set of all sets), but there are several instances of infinite expressions that are well defined. Examples of well-defined infinite expressions include〔 〕〔 〕 infinite sums, whether expressed using summation notation or as an infinite series, such as :$\backslash sum\_^\backslash infty\; a\_n\; =\; a\_0\; +\; a\_1\; +\; a\_2\; +\; \backslash cdots\; \backslash ,;$ infinite products, whether expressed using product notation or expanded, such as :$\backslash prod\_^\backslash infty\; b\_n\; =\; b\_0\; \backslash times\; b\_1\; \backslash times\; b\_2\; \backslash times\; \backslash cdots\; \backslash ,$ infinite nested radicals, such as : $\backslash sqrt\; \backslash ,$ infinite power towers, such as : $\backslash sqrt^^\}\}\; \backslash ,$ and infinite continued fractions, whether expressed using Gauss's Kettenbruch notation or expanded, such as :$c\_0\; +\; \backslash underset\}\; \backslash ,\; \backslash frac\; =\; c\_0\; +\; \backslash cfrac\}\}\}\backslash ,$ In infinitary logic, one can use infinite conjunctions and infinite disjunctions. Even for well-defined infinite expressions, the ''value'' of the infinite expression may be ambiguous or not well defined; for instance, there are multiple summation rules available for assigning values to series, and the same series may have different values according to different summation rules if the series is not absolutely convergent. ==From the hyperreal viewpoint== From the point of view of the hyperreals, such an infinite expression $E\_\backslash infty$ is obtained in every case from the sequence $\backslash langle\; E\_n\; :\; n\; \backslash in\; \backslash mathbb\backslash rangle$ of finite expressions, by evaluating the sequence at a hypernatural value $n=H$ of the index ''n'', and applying the standard part, so that $E\_\backslash infty=\backslash text(E\_H)$. 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 expression (mathematics)」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.