or an integral
is said to converge absolutely if the series or integral of the corresponding absolute value is finite, i.e.
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.
Series or integrals that converge but do not converge absolutely are said to converge conditionally.