Absolute convergence

In mathematics, a series

<math>\sum_{n=1}^\infty a_n<math>

or an integral

<math>\int_A f(x)\,dx<math>

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

<math>\sum_{n=1}^\infty \left|a_n\right|<\infty<math>

or, respectively,

<math>\int_A \left|f(x)\right|\,dx<\infty.<math>

Absolute convergence entails that rearrangement of the series

<math>\sum_{n=1}^\infty a_{\sigma(n)}<math>

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.