Bound for summation

Main Question or Discussion Point

I am looking for a bound for the following expression

$$S=\sum_{n=1}^N n^k e^{-an}$$
where a>0 and k=1, 2, 3, or 4, apart from the obvious one:

$$S\le \frac{n+1}{2} \sum_{n=1}^N e^{-an} = \frac{n+1}{2} \frac{1-e^{-Na}}{e^a-1}$$

Last edited:

$$S\le \int_1^{N+1} x^k e^{-ax} dx$$