I hope this question is not too simple, but I would like to know the asymptotic behaviour of the following function $f: \mathbb{N}^{+} \rightarrow \mathbb{Q}$ where $$ f(n) = \sum_{i=1}^{n} \frac{i^n}{n^{4i}} $$ Any references, pointers, or answers would be most appreciated.

From the comments so far (including mine above) it follows that $$ f(n) = \sum_{i=1}^{n} \frac{i^n}{n^{4i}}=\frac{n!}{(4\ln n)^{n+1}}\left(1+O(n^{1/2}\ln n)\right). $$ 


$f(n) = n^n 4^{4n} (1 + O(n^{4}))$. The sum is strongly dominated by its last term. I hope this isn't homework. Apologies. I misread the question exact as Benoît suggests. 

