Bound for S: Sum of n^k e^(-an)

[bruno67]

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:

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

 

[bruno67]

I got it. S(k) is bound by the integral

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