# series

Given a sequence of numbers (real or complex) $\{{a}_{n}\}$ we define a sequence of partial sums $\{{S}_{N}\}$, where ${S}_{N}={\sum}_{n=1}^{N}{a}_{n}$. This sequence is called the series with terms ${a}_{n}$. We define the sum of the series ${\sum}_{n=1}^{\mathrm{\infty}}{a}_{n}$ to be the limit of these partial sums. More precisely

$$\sum _{n=1}^{\mathrm{\infty}}{a}_{n}=\underset{N\to \mathrm{\infty}}{lim}{S}_{n}=\underset{N\to \mathrm{\infty}}{lim}\sum _{n=1}^{N}{a}_{n}.$$ |

In a context where this distinction does not matter much (this is usually the case) one identifies a series with its sum, if the latter exists.

Traditionally, as above, series are infinite sums of real numbers. However,
the formal constraints on the terms $\{{a}_{n}\}$ are much less strict. We need
only be able to add the terms and take the limit of partial sums. So in full
generality the terms could be complex numbers^{} or even elements of certain rings,
fields, and vector spaces.

Title | series |
---|---|

Canonical name | Series |

Date of creation | 2013-03-22 12:41:35 |

Last modified on | 2013-03-22 12:41:35 |

Owner | mathwizard (128) |

Last modified by | mathwizard (128) |

Numerical id | 7 |

Author | mathwizard (128) |

Entry type | Definition |

Classification | msc 40-00 |

Related topic | AbsoluteConvergence |

Related topic | HarmonicNumber |

Related topic | CompleteUltrametricField |

Related topic | Summation |

Related topic | PrimeHarmonicSeries |