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 $\{\log p_{n}\}$ ( $\log$ is restricted to its principal branch) converges to a finite limit. By the Cauchy criterion, this happens iff for every $\epsilon^{\prime}>0$ there exist $N$ such that $\left|\log p_{n+k}-\log p_{n}\right|<\epsilon^{\prime}$ for all $n>N$ and all $k=1,2,\ldots$ , i.e, iff

$\left|\log\frac{p_{n+k}}{p_{n}}\right|=\left|\log u_{n+1}u_{n+2}\cdots u_{n+k}% \right|<\epsilon^{\prime};$

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

$\left|u_{n+1}u_{n+2}\cdots u_{n+k}-1\right|<\epsilon$

for $n$ greater than $N$ and $k=1,2,\ldots$ ∎

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