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# convergent series

A series $\sum a_{n}$ is said to be *convergent*
if the sequence of partial sums $\sum_{{i=1}}^{n}a_{i}$ is convergent.
A series that is not convergent is said to be *divergent*.

A series $\sum a_{n}$ is said to be *absolutely convergent*
if $\sum|a_{n}|$ is convergent.

When the terms of the series live in $\mathbb{R}^{n}$, an equivalent condition for absolute convergence of the series is that all possible series obtained by rearrangements of the terms are also convergent. (This is not true in arbitrary metric spaces.)

It can be shown that absolute convergence implies convergence. A series that converges, but is not absolutely convergent, is called conditionally convergent.

Defines:

absolute convergence, conditional convergence, absolutely convergent, conditionally convergent, converges absolutely, convergent, divergent, divergent series

Related:

Series, HarmonicNumber, ConvergesUniformly, SumOfSeriesDependsOnOrder, UncoditionalConvergence, WeierstrassMTest, DeterminingSeriesConvergence

Type of Math Object:

Definition

Major Section:

Reference

Groups audience:

## Mathematics Subject Classification

26A06*no label found*40A05

*no label found*

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## Attached Articles

manipulating convergent series by pahio

slower convergent series by pahio

slower divergent series by pahio

double series by PrimeFan

Riemann's theorem on rearrangements by Gorkem

convergence of complex term series by pahio

conditionally convergent real series by pahio

finite changes in convergent series by pahio

slower convergent series by pahio

slower divergent series by pahio

double series by PrimeFan

Riemann's theorem on rearrangements by Gorkem

convergence of complex term series by pahio

conditionally convergent real series by pahio

finite changes in convergent series by pahio