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Compute the Lebesgue integral $\int_0^{\infty} \frac{x}{e^x -1}dx$.

I think I need to use the Dominated Convergence Theorem or the Beppo Levi Theorem to show this, but I don't really know what I should do with the function. How can I compute this integral? I would greatly appreciate any help.

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  • $\begingroup$ Just because this question came after learning these. But I'm completely lost on how to evaluate this. $\endgroup$ Apr 23, 2016 at 7:15

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Hint: $$\frac{x}{e^x-1}=\sum_{n=1}^\infty xe^{-nx}$$

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  • $\begingroup$ Can you tell me how to get this equality? $\endgroup$ Apr 23, 2016 at 7:21
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    $\begingroup$ @takecare $e^{-nx}=(e^{-x})^n$. This is a geometric series. $\endgroup$
    – Wojowu
    Apr 23, 2016 at 7:23

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