# proof of convergence condition of infinite product

proof of theorem of convergence of infinite productFernando Sanz Gamiz

###### Proof.

Let ${p}_{n}={\prod}_{i=1}^{n}{u}_{i}$. We have to study the convergence of the
sequence $\{{p}_{n}\}$. The sequence $\{{p}_{n}\}$ converges to a not null limit iff
$\{\mathrm{log}{p}_{n}\}$ ($\mathrm{log}$ is restricted to its principal branch^{}) converges
to a finite limit. By the Cauchy criterion, this happens iff for
every ${\u03f5}^{\prime}>0$ there exist $N$ such that $$ for all $n>N$ and all
$k=1,2,\mathrm{\dots}$, i.e, iff

$$ |

as $\mathrm{log}(z)$ is an injective function and continuous at $z=1$ and $\mathrm{log}(1)=0$ this will happen iff for every $\u03f5>0$

$$ |

for $n$ greater than $N$ and $k=1,2,\mathrm{\dots}$ ∎

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

Canonical name | ProofOfConvergenceConditionOfInfiniteProduct |

Date of creation | 2013-03-22 17:22:27 |

Last modified on | 2013-03-22 17:22:27 |

Owner | fernsanz (8869) |

Last modified by | fernsanz (8869) |

Numerical id | 5 |

Author | fernsanz (8869) |

Entry type | Proof |

Classification | msc 30E20 |