Editing Absolute convergence (section)

{{Short description|Mode of convergence of an infinite series}}
In [[mathematics]], an [[Series (mathematics)|infinite series]] of numbers is said to '''converge absolutely''' (or to be '''absolutely convergent''') if the sum of the [[absolute value]]s of the summands is finite<!-- don't link to [[finite set]], please -->.  More precisely, a [[Real number|real]] or [[Complex number|complex]] series <math>\textstyle\sum_{n=0}^\infty a_n</math> is said to '''converge absolutely''' if <math>\textstyle\sum_{n=0}^\infty \left|a_n\right| = L</math> for some real number <math>\textstyle L.</math> Similarly, an [[improper integral]] of a [[function (mathematics)|function]], <math>\textstyle\int_0^\infty f(x)\,dx,</math> is said to converge absolutely if the integral of the absolute value of the integrand is finite—that is, if <math>\textstyle\int_0^\infty |f(x)|dx = L.</math>  A convergent series that is not absolutely convergent is called [[Conditional convergence|conditionally convergent]].

Absolute convergence is important for the study of infinite series, because its definition guarantees that a series will have some "nice" behaviors of finite sums that not all convergent series possess. For instance, rearrangements do not change the value of the sum, which is not necessarily true for conditionally convergent series.