# 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 ProofOfConvergenceConditionOfInfiniteProduct 2013-03-22 17:22:27 2013-03-22 17:22:27 fernsanz (8869) fernsanz (8869) 5 fernsanz (8869) Proof msc 30E20