# Math Help - Series divergence

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?

Please help!
Thanks

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

The last term of this product will be infinite and there will be only 999 terms less than one, none of which are less than 0.001. So you are multiplying an infinite number by a finite one and so will get infinity.

There may well be a cleaner way to express this.