PlanetMath: convergent series

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convergent series

(Definition)

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 $\sum \vert a_n\vert$ 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.

"convergent series" is owned by yark. [ full author list (3) | owner history (3) ]

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Also defines:

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

Attachments:: manipulating convergent series (Theorem) by pahio
slower convergent series (Theorem) by pahio
slower divergent series (Theorem) by pahio
double series (Theorem) by PrimeFan
Riemann's theorem on rearrangements (Theorem) by Gorkem

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Cross-references: sum, converges, implies, metric spaces, equivalent, terms, partial sums, sequence, series
There are 129 references to this entry.

This is version 9 of convergent series, born on 2002-02-20, modified 2008-04-18.
Object id is 2311, canonical name is AbsoluteConvergence.
Accessed 28927 times total.

Classification:

AMS MSC:	40A05 (Sequences, series, summability :: Convergence and divergence of infinite limiting processes :: Convergence and divergence of series and sequences)
	26A06 (Real functions :: Functions of one variable :: One-variable calculus)

Pending Errata and Addenda

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