# Abel’s limit theorem

Suppose that $\sum {a}_{n}{x}^{n}$ has a radius of convergence^{} $r$ and that $\sum {a}_{n}{r}^{n}$ is convergent. Then

$$\underset{x\to {r}^{-}}{lim}\sum {a}_{n}{x}^{n}=\sum {a}_{n}{r}^{n}=\sum (\underset{x\to {r}^{-}}{lim}{a}_{n}{x}^{n})$$ |

Title | Abel’s limit theorem |
---|---|

Canonical name | AbelsLimitTheorem |

Date of creation | 2013-03-22 12:57:49 |

Last modified on | 2013-03-22 12:57:49 |

Owner | CWoo (3771) |

Last modified by | CWoo (3771) |

Numerical id | 5 |

Author | CWoo (3771) |

Entry type | Theorem |

Classification | msc 40A30 |

Related topic | PowerSeries |

Related topic | AbelsMultiplicationRuleForSeries |

Related topic | AbelSummability |

Related topic | NielsHenrikAbel |