Leading coefficient of a sum of thousandth powers

The sum $\sum_{k=1}^{n}k^{1000}$ equals a polynomial in $n$ whose highest-degree term is $a\,n^{1001}$. Find $a$.

$a=\tfrac{1}{1001}$. Dividing by $n^{1001}$ gives the Riemann sum $\tfrac1n\sum(k/n)^{1000}\to\int_0^1x^{1000}dx=\tfrac{1}{1001}$, which must equal the leading coefficient.