convergent series

A series an is said to be convergentMathworldPlanetmathPlanetmath if the sequence of partial sums i=1nai is convergent. A series that is not convergent is said to be divergent.

A series an is said to be absolutely convergent if |an| is convergent.

When the terms of the series live in n, an equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath 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 convergesPlanetmathPlanetmath, but is not absolutely convergent, is called conditionally convergent.

Let an be an absolutely convergent series, and bn be a conditionally convergent series. Then any rearrangement of an is convergent to the same sum. It is a result due to Riemann that bn can be rearranged to converge to any sum, or not converge at all.

Title convergent seriesMathworldPlanetmath
Canonical name ConvergentSeries
Date of creation 2013-03-22 12:24:51
Last modified on 2013-03-22 12:24:51
Owner yark (2760)
Last modified by yark (2760)
Numerical id 12
Author yark (2760)
Entry type Definition
Classification msc 26A06
Classification msc 40A05
Related topic Series
Related topic HarmonicNumber
Related topic ConvergesUniformly
Related topic SumOfSeriesDependsOnOrder
Related topic UncoditionalConvergence
Related topic WeierstrassMTest
Related topic DeterminingSeriesConvergence
Defines absolute convergence
Defines conditional convergence
Defines absolutely convergent
Defines conditionally convergent
Defines converges absolutely
Defines convergent
Defines divergent
Defines divergent series