Absolute convergence
From Academic Kids
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.de:Absolute Konvergenz es:Convergencia absoluta
