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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 is said to converge absolutely if for some real number . Similarly, an improper integral of a function, , is said to converge absolutely if the integral of the absolute value of the integrand is finite—that is, if 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 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 on abelian group ''G'' (written additively, with identity element 0) such that: # The norm of the identity element of ''G'' is zero: # For every ''x'' in ''G'', implies # For every ''x'' in ''G'', # For every ''x'', ''y'' in ''G'', In this case, the function 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
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