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Bound for summation

  1. Dec 5, 2011 #1
    I am looking for a bound for the following expression

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

    [tex]S\le \frac{n+1}{2} \sum_{n=1}^N e^{-an} = \frac{n+1}{2}
    \frac{1-e^{-Na}}{e^a-1}[/tex]
     
    Last edited: Dec 5, 2011
  2. jcsd
  3. Dec 5, 2011 #2
    I got it. S(k) is bound by the integral

    [tex]S\le \int_1^{N+1} x^k e^{-ax} dx[/tex]
     
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