# Absolute convergence

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In mathematics, a series

$\sum _{n=1}^{\infty }a_{n}$ or an integral

$\int _{A}f(x)\,dx$ is said to converge absolutely if the series or integral of the corresponding absolute value is finite, i.e.

$\sum _{n=1}^{\infty }\left|a_{n}\right|<\infty$ or, respectively,

$\int _{A}\left|f(x)\right|\,dx<\infty .$ If $a_{n}$ is a complex number, this theorem can be imagined as follows: the sum of all $a_{k}$ is a vector addition path through the complex plane. If the length of the path, that is the sum of all the lengths of the parts $|a_{k}|$ , is finite, the end point has to be a finite distance from the origin.

Absolute convergence entails that rearrangement of the series

$\sum _{n=1}^{\infty }a_{\sigma (n)}$ where σ is a permutation of the natural numbers, does not alter the sum to which the series converges. Similar results apply to integrals. See Cauchy principal value and an elegant rearrangement of a conditionally convergent iterated integral.

In the light of Lebesgue's theory of integration, sums may be treated as special cases of integrals, rather than as a separate case.

Series or integrals that converge but do not converge absolutely are said to converge conditionally.