# comparison test

The series

$$\sum _{i=0}^{\mathrm{\infty}}{a}_{i}$$ |

with real ${a}_{i}$ is absolutely convergent if there is a sequence ${({b}_{n})}_{n\in \mathbb{N}}$ with positive real ${b}_{n}$ such that

$$\sum _{i=0}^{\mathrm{\infty}}{b}_{i}$$ |

is and for all sufficiently large $k$ holds $|{a}_{k}|\le {b}_{k}$.

Also, the series $\sum {a}_{i}$ is divergent if there is a sequence $({b}_{n})$ with positive real ${b}_{n}$, so that $\sum {b}_{i}$ is divergent and ${a}_{k}\ge {b}_{k}$ for all sufficiently large $k$.

Title | comparison test^{} |
---|---|

Canonical name | ComparisonTest |

Date of creation | 2013-03-22 13:21:48 |

Last modified on | 2013-03-22 13:21:48 |

Owner | mathwizard (128) |

Last modified by | mathwizard (128) |

Numerical id | 4 |

Author | mathwizard (128) |

Entry type | Theorem |

Classification | msc 40A05 |