# 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.

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

 Title convergent series 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