Are there some (bivariate) closed form formulas for the asymptotic behaviour of the sum:
$$\sum_{k=1}^{n} k^d,$$
where $n$ and $d$ are large integers? I am especially interested in a lower bound of the form $\Omega(f(n,d))$. I am aware there is an exact formula as a polynomial involving the Bernouilli numbers, and the asymptotic behavior has been discussed in some other question when $d$ is fixed, but here I consider both $n$ and $d$ as unbounded.
