# proof of absolute convergence theorem

Suppose that $\sum {a}_{n}$ is absolutely convergent, i.e., that $\sum |{a}_{n}|$ is convergent. First of all, notice that

$$0\le {a}_{n}+|{a}_{n}|\le 2|{a}_{n}|,$$ |

and since the series $\sum ({a}_{n}+|{a}_{n}|)$ has non-negative terms it can be compared with $\sum 2|{a}_{n}|=2\sum |{a}_{n}|$ and hence converges.

On the other hand

$$\sum _{n=1}^{N}{a}_{n}=\sum _{n=1}^{N}({a}_{n}+|{a}_{n}|)-\sum _{n=1}^{N}|{a}_{n}|.$$ |

Since both the partial sums on the right hand side are convergent, the partial sum on the left hand side is also convergent. So, the series $\sum {a}_{n}$ is convergent.

Title | proof of absolute convergence theorem |
---|---|

Canonical name | ProofOfAbsoluteConvergenceTheorem |

Date of creation | 2013-03-22 13:41:52 |

Last modified on | 2013-03-22 13:41:52 |

Owner | paolini (1187) |

Last modified by | paolini (1187) |

Numerical id | 6 |

Author | paolini (1187) |

Entry type | Proof |

Classification | msc 40A05 |