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. Because of Lebesgue's theory of integration, sums may be regarded as integrals rather than as a separate case. Series or integrals that converge but do not converge absolutely are said to converge conditionally.

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