# absolute convergence of infinite product and series

Theorem. The infinite product

$$\prod _{n=1}^{\mathrm{\infty}}(1+{c}_{n})$$ |

converges absolutely (http://planetmath.org/AbsoluteConvergenceOfInfiniteProduct) if and only if the series

$$\sum _{n=1}^{\mathrm{\infty}}{c}_{n}$$ |

with complex ${c}_{n}$ converges absolutely (http://planetmath.org/AbsoluteConvergence).

Proof. The theorem follows directly from the theorems of the entries absolutely convergent infinite product converges and infinite product of sums $1+{a}_{i}$ (http://planetmath.org/InfiniteProductOfSums1a_i).

Title | absolute convergence of infinite product and series |
---|---|

Canonical name | AbsoluteConvergenceOfInfiniteProductAndSeries |

Date of creation | 2013-03-22 18:41:18 |

Last modified on | 2013-03-22 18:41:18 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 6 |

Author | pahio (2872) |

Entry type | Axiom |

Classification | msc 30E20 |

Related topic | AbsoluteConvergentInfiniteProductConverges |

Related topic | InfiniteProductOfSums1a_i |

Related topic | OrderOfFactorsInInfiniteProduct |