I'm working on a convergence problem that's giving me trouble. I'll list the steps I've made so far.
Given the following series determine if it is convergent or divergent: $\sum_{n=1}^{\infty}\frac{n!\cdot x^n}{n^n}, \text{where } x > 0.$
When I first saw this problem I thought to use the root test so I attempted to preform the following calculation: $\lim_{n\to\infty} \sqrt[n]{\left| \frac{n!\cdot x^n}{n^n} \right|}.$
But here is where I'm not sure how to move forward. I'm basically unsure if we can distribute the $\frac{1}{n}$ exponent to $n!$ to generate something like this ${(n!)}^{1/n} \cdot x$ as the numerator (which would go to $x$ as $n \longrightarrow \infty$).