Convergence of Infinite Product for all Finite x

Question

I am having some trouble showing that the $\prod_{n=1}^\infty (1+x/n)e^{-x/n}$ converges for all finite $x$ but $0$. So far, I have shown that if we take the log, it reduces to $\sum_{n=1}^\infty (\ln(1+x/n)-{x/n})$. I wanted to use the Taylor Series for $\ln(1+x)$ but its radius of convergence is $1$ so I know that doesn't work. How do I proceed?

Answer 1

For any fixed $x$, $\ln(1 + x/n) - x/n = O(1/n^2)$ as $n \to \infty$, and $\sum_n 1/n^2$ converges. Use Limit Comparison Test.