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Math Help - Integral of a floor function

  1. #1
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    Integral of a floor function

    Greetings.

    A bit of help would be much appreciated on the following integral:


    \int_{0}^{\infty} \lfloor x \rfloor  e^{-x} dx

    Edit:

    Ah this is just a geometric series:  \int_{n-1}^{n} n e^{-x} dx
    Last edited by mr fantastic; August 28th 2009 at 06:58 PM. Reason: Fixed latex
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  2. #2
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    Krizalid's Avatar
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    Note \int_{j}^{j+1}{e^{-x}\,dx}=(e-1)e^{-j-1}, so \int_{0}^{\infty }{\left\lfloor x \right\rfloor e^{-x}\,dx}=\sum\limits_{j=0}^{\infty }{\int_{j}^{j+1}{\left\lfloor x \right\rfloor e^{-x}\,dx}}=(e-1)\sum\limits_{j=0}^{\infty }{je^{-j-1}}.

    Computation of that series will give ya the answer.
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  3. #3
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    Good catch it looks like we cross posted.
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