For a finite case we have $\lim\limits_{n\rightarrow\infty}f(n)\cdot g(n) =\lim\limits_{n\rightarrow\infty}f(n)\cdot\lim\limits_{n\rightarrow\infty}g(n)$ however when is it possible to interchange the folowwing?
$\lim\limits_{n\rightarrow\infty}\prod\limits_{i=1}^n f(i)$
A specific problem I have is $\lim\limits_{n\rightarrow\infty}\prod\limits_{i=1}^n\frac{2i-1}{3i-2}$. I tried taking the natural logarithm and this results in $\lim\limits_{n\rightarrow\infty}\sum\limits_{i=1}^n\ln\left(\frac{2i-1}{3i-2}\right)$.
If I pass the limit then Ill be multiplying 2/3 infinitely often resulting in zero. Is this logic sound? Come to think of it, $\frac{2n-1}{3n-2}<1$ for every $n>1$ so the product is zero. No need to pass the limit. However the original question with $f(n)$ is still interesting.