modern analysis: integrals and continuity Let $$f(x) = \sum_1 ^\infty n*e^{-nx}$$ Where is $f$ continuous? Compute  $$\int_1^2f(x) dx$$
I am having trouble proving where $f$ is continuous. For the second part, so far I have been able to compute the derivative.. although I basically had to move the summation from inside the integral to outside the integral and I am not sure why I am allowed to do that.

When the real part of $x$ is greater than $0$, $f(x)=\frac{e^x}{(e^x-1)^2}$. Clearly this is continuous so simply show the relation.
When the real part is less than $0$, use the limit test.
