# Absolute convergence

In mathematics, a series

${\displaystyle \sum _{n=1}^{\infty }a_{n}}$

or an integral

${\displaystyle \int _{A}f(x)\,dx}$

is said to converge absolutely if the series or integral of the corresponding absolute value is finite, i.e.

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

or, respectively,

${\displaystyle \int _{A}\left|f(x)\right|\,dx<\infty .}$

If ${\displaystyle a_{n}}$ is a complex number, this theorem can be imagined as follows: the sum of all ${\displaystyle a_{k}}$ 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 ${\displaystyle |a_{k}|}$, is finite, the end point has to be a finite distance from the origin.

Absolute convergence entails that rearrangement of the series

${\displaystyle \sum _{n=1}^{\infty }a_{\sigma (n)}}$

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.