# proof of Abel’s test for convergence

Let $b$ be the limit of $\{{b}_{n}\}$ and let ${d}_{n}={b}_{n}-b$ when $\{{b}_{n}\}$ is decreasing and ${d}_{n}=b-{b}_{n}$ when $\{{b}_{n}\}$ is increasing. By Dirichlet’s convergence test, $\sum {a}_{n}{d}_{n}$ is convergent^{} and so is $\sum {a}_{n}{b}_{n}=\sum {a}_{n}(b\pm {d}_{n})=b\sum {a}_{n}\pm \sum {a}_{n}{d}_{n}$.

Title | proof of Abel’s test for convergence |
---|---|

Canonical name | ProofOfAbelsTestForConvergence |

Date of creation | 2013-03-22 13:19:56 |

Last modified on | 2013-03-22 13:19:56 |

Owner | lieven (1075) |

Last modified by | lieven (1075) |

Numerical id | 4 |

Author | lieven (1075) |

Entry type | Proof |

Classification | msc 40A05 |