calculus - How to evaluate $lim_{ntoinfty}n!/n^{k}$

I do not know how to go about finding the limit of this sequence. I know it diverges to infinity yet I can't find terms to use the squeeze lemma effectively.

$a_n = \frac{n!}{n^{1000}}$

$\lim_{n\rightarrow\infty} \frac{n!}{n^{1000}}$

I know that when $n>1000$, the numerator will "run out" of denominators and the expression will grow, yet I don't know how to formally prove this divergence.

In addition to this, since it is a sequence, no series laws can be applied.

Anyone have an idea on how to approach this type of problem?

Answer

I will first assume that $k$ is a positive integer. Observe that

$$\frac{n!}{n^k}=\frac{n}{n}\frac{n-1}{n}\cdots\frac{n-k+1}{n}(n-k)!\geq \Big(1-\frac{k-1}{n}\Big)^k(n-k)!\geq 2^{-k}(n-k)!$$ for all sufficiently large $n$. And $$\lim_{n\to\infty}2^{-k}(n-k)!=\infty$$ for $k$ fixed, so the original sequence diverges as well.

If $k$ is a positive real number but not an integer, then if $j=\lceil k\rceil$ then

$$\frac{n!}{n^k}\geq \frac{n!}{n^j}$$ so we can use the above argument.