Absolute convergence

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In mathematics, a series

or an integral

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

or, respectively,

If is a complex number, this theorem can be imagined as follows: the sum of all 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 , is finite, the end point has to be a finite distance from the origin.

Absolute convergence entails that rearrangement of the series

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.

de:Absolute Konvergenz es:Convergencia absoluta