Limit Test on Series: Summation from n=0 to Infinity of n!/1000^n

[cue928]

Series: summation from n=0 to infinity of n!/1000^n

I can look at that and see the limit is not going to zero but how do you show that? Also, were it not in the first section of the book (i.e. before the ratio test), I would have tried to use the ratio test on it - is that acceptable to do? I got infinity for the answer under the ratio test, btw.

 

[Jyupiter]

Also, were it not in the first section of the book (i.e. before the ratio test), I would have tried to use the ratio test on it - is that acceptable to do?

Sure, why not?

I can look at that and see the limit is not going to zero but how do you show that?

Do you think $n!>1000^{n}$ for some large n to be true? If so, could you manipulate the rational function using this knowledge?