1. ## Series divergence

I know that the series
Σ n! / 1000^n
diverges because n! eventually becomes much bigger than 1000^n. But I need to prove that the limit diverges to infinity. We haven't covered the ratio test yet but I know once I prove the limit is infinity I can use the divergence test. What is the best way to show the limit is in fact, infinity?

2. $\displaystyle \frac {n!}{1000^n} = \prod_{i=1}^{n} \frac {i}{1000}$