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

Question

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.

Answer 1

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