math lessons - Absolute convergence

$\sum_{n=1}^\infty a_n$

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,

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

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.

Categories: Mathematical analysis | Infinite series | Integration

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