# convergence condition of infinite product

Let us think the sequence^{} ${u}_{1},{u}_{1}{u}_{2},{u}_{1}{u}_{2}{u}_{3},\mathrm{\dots}$ In the complex analysis, one often uses the definition of the convergence of an infinite product $\prod _{k=1}^{\mathrm{\infty}}}{u}_{k$ where the case $\underset{k\to \mathrm{\infty}}{lim}{u}_{1}{u}_{2}\mathrm{\dots}{u}_{k}=0$ is excluded. Then one has the

###### Theorem.

The infinite product $\prod _{k=1}^{\mathrm{\infty}}}{u}_{k$ of the non-zero complex numbers^{} ${u}_{1}$, ${u}_{2}$, … is convergent iff for every positive number $\epsilon $ there exists a positive number ${n}_{\epsilon}$ such that the condition

$$ |

is true as soon as $n\geqq {n}_{\epsilon}$.

Corollary. If the infinite product converges, then we necessarily have $\underset{k\to \mathrm{\infty}}{lim}{u}_{k}=1$. (Cf. the necessary condition of convergence of series.)

When the infinite product converges, we say that the value of the infinite product is equal to $\underset{k\to \mathrm{\infty}}{lim}{u}_{1}{u}_{2}\mathrm{\dots}{u}_{k}$.

Title | convergence condition of infinite product |
---|---|

Canonical name | ConvergenceConditionOfInfiniteProduct |

Date of creation | 2013-03-22 14:37:22 |

Last modified on | 2013-03-22 14:37:22 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 16 |

Author | pahio (2872) |

Entry type | Theorem |

Classification | msc 30E20 |

Related topic | OrderOfFactorsInInfiniteProduct |

Related topic | NecessaryConditionOfConvergence |

Defines | infinite product |

Defines | value of infinite product |