Absolute convergence

Mode of convergence of an infinite series

In mathematics, an infinite series of numbers is said to converge absolutely (or to be absolutely convergent) if the sum of the absolute values of the summands is finite. More precisely, a real or complex series ∑ n = 0 ∞ a n {\displaystyle \textstyle \sum _{n=0}^{\infty }a_{n}} is said to converge absolutely if ∑ n = 0 ∞ | a n | = L {\displaystyle \textstyle \sum _{n=0}^{\infty }\left|a_{n}\right|=L} for some real number L . {\displaystyle \textstyle L.} Similarly, an improper integral of a function, ∫ 0 ∞ f ( x) d x , {\displaystyle \textstyle \int _{0}^{\infty }f(x)\,dx,} is said to converge absolutely if the integral of the absolute value of the integrand is finite—that is, if ∫ 0 ∞ | f ( x) | d x = L .

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