Absolute convergence

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In mathematics, a series

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

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,

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

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

Absolute convergence entails that rearrangement of the series

$\sum_{n=1}^\infty a_{\sigma(n)}$