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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 Absolute convergence ： ウィキペディア英語版 Absolute convergence In mathematics, an infinite series of numbers is said to converge absolutely (or to be absolutely convergent) if the sum of the absolute value of the summand is finite. More precisely, a real or complex series $\backslash textstyle\backslash sum\_^\backslash infty\; a\_n$ is said to converge absolutely if $\backslash textstyle\backslash sum\_^\backslash infty\; \backslash left|a\_n\backslash right|\; =\; L$ for some real number $\backslash textstyle\; L$. Similarly, an improper integral of a function, $\backslash textstyle\backslash int\_0^\backslash infty\; f(x)\backslash ,dx$, is said to converge absolutely if the integral of the absolute value of the integrand is finite—that is, if $\backslash textstyle\backslash int\_0^\backslash infty\; \backslash left|f(x)\backslash right|dx\; =\; L.$ Absolute convergence is important for the study of infinite series because its definition is strong enough to have properties of finite sums that not all convergent series possess, yet is broad enough to occur commonly. (A convergent series that is not absolutely convergent is called conditionally convergent.) ==Background== One may study the convergence of series $\backslash sum\_^\; a\_n$ whose terms ''an'' are elements of an arbitrary abelian topological group. The notion of absolute convergence requires more structure, namely a norm, which is a real-valued function $\backslash |\backslash cdot\backslash |:\; G\; \backslash to\; \backslash mathbb$ on abelian group ''G'' (written additively, with identity element 0) such that: # The norm of the identity element of ''G'' is zero: $\backslash |0\backslash |\; =\; 0.$ # For every ''x'' in ''G'', $\backslash |x\backslash |\; =\; 0$ implies $x\; =\; 0.$ # For every ''x'' in ''G'', $\backslash |-x\backslash |\; =\; \backslash |x\backslash |.$ # For every ''x'', ''y'' in ''G'', $\backslash |x+y\backslash |\; \backslash leq\; \backslash |x\backslash |\; +\; \backslash |y\backslash |.$ In this case, the function $d(x,y)\; =\; \backslash |x-y\backslash |$ induces on ''G'' the structure of a metric space (a type of topology). We can therefore consider ''G''-valued series and define such a series to be absolutely convergent if In particular, these statements apply using the norm |''x''| (absolute value) in the space of real numbers or complex numbers. 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「Absolute convergence」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.