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 thesequence $\\{p_{n}\\}$ . The sequence $\\{p_{n}\\}$ converges to a not null limit iff $\\{\\log p_{n}\\}$ ( $\\log$ is restricted to its principal branch) convergesto a finite limit. By the Cauchy criterion, this happens iff forevery $\\epsilon^{\\prime}>0$ 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 $n$ greaterthan $N$ and $k=1,2,\\ldots$ ∎