Absolute convergence

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In mathematics, a series (or sometimes also an integral) of numbers is said to converge absolutely if the sum (or integral) of the absolute value of the summand or integrand is finite. More precisely, a real or complex series \sum_{n=0}^\infty a_n is said to converge absolutely if \sum_{n=0}^\infty \left|a_n\right| < \infty.

Absolute convergence is important to the study of infinite series because its definition is strong enough to have properties of finite sums that not all convergent series possess, yet is broad enough to occur commonly.

Contents

Background

One may study the convergence of series \sum_{n=0}^{\infty} a_n whose terms an are elements of an arbitrary abelian topological group. The notion of absolute convergence requires more structure, namely a norm:

A norm on an abelian group G (written additively, with identity element 0) is a real-valued function x \mapsto \|x\| on G such that:

Then the function d(x,y) = \|x-y\| induces on G the structure of a metric space (in particular, a topology). We can therefore consider G-valued series and define such a series to be absolutely convergent if \sum_{n=0}^{\infty} \|a_n\| < \infty.

Relation to convergence

If the metric d on G is complete, then every absolutely convergent series is convergent. The proof is the same as for complex-valued series: use the completeness to derive the Cauchy criterion for convergence—a series is convergent if and only if its tails can be made arbitrarily small in norm—and apply the triangle inequality.

In particular, for series with values in any Banach space, absolute convergence implies convergence. The converse is also true: if absolute convergence implies convergence in a normed space, then the space is a Banach space.

If a series is convergent but not absolutely convergent, it is called conditionally convergent. An example of a conditionally convergent series is the alternating harmonic series. Many standard tests for divergence and convergence, most notably including the ratio test and the root test, demonstrate absolute convergence. This is because a power series is absolutely convergent on the interior of its disk of convergence.

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