How can we approximate (or bound from below) $$(1-\frac{n^a}{n})^{n^b},$$ where $n^{a+b}>n$, and $n$ is large?
When $n$ becomes large, I guess this should behave like $e^{-n^{a+b-1}}$, so very close to zero. However, I would like to have a lower bound, or an estimate of the error term, such as $$(1-\frac{n^a}{n})^{n^b}=O(f(n,a,b)),$$ where preferably, $f(n,a,b)$ is relatively small. Do any of you know such bounds?