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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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## Recent Activity

Apr 15

new question: Prove a formula is part of the Gentzen System by LadyAnne

Mar 30

new question: A problem about Euler's totient function by mbhatia

new problem: Problem: Show that phi(a^n-1), (where phi is the Euler totient function), is divisible by n for any natural number n and any natural number a >1. by mbhatia

new problem: MSC browser just displays "No articles found. Up to ." by jaimeglz

Mar 26

new correction: Misspelled name by DavidSteinsaltz

Mar 21

new correction: underline-typo by Filipe

Mar 19

new correction: cocycle pro cocyle by pahio

Mar 7

new image: plot W(t) = P(waiting time <= t) (2nd attempt) by robert_dodier

new image: expected waiting time by robert_dodier

new image: plot W(t) = P(waiting time <= t) by robert_dodier

new question: Prove a formula is part of the Gentzen System by LadyAnne

Mar 30

new question: A problem about Euler's totient function by mbhatia

new problem: Problem: Show that phi(a^n-1), (where phi is the Euler totient function), is divisible by n for any natural number n and any natural number a >1. by mbhatia

new problem: MSC browser just displays "No articles found. Up to ." by jaimeglz

Mar 26

new correction: Misspelled name by DavidSteinsaltz

Mar 21

new correction: underline-typo by Filipe

Mar 19

new correction: cocycle pro cocyle by pahio

Mar 7

new image: plot W(t) = P(waiting time <= t) (2nd attempt) by robert_dodier

new image: expected waiting time by robert_dodier

new image: plot W(t) = P(waiting time <= t) by robert_dodier

## 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