# 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 |