TITLE: If $f(x) = e^{-x}+2e^{-2x}+3e^{-3x}+\cdots$ then find $\int_{\ln2}^{\ln3}f(x)dx$
QUESTION [8 upvotes]: Solve the following 
If $f(x) = e^{-x}+2e^{-2x}+3e^{-3x}+\cdots $
Then find $\int_{\ln2}^{\ln3}f(x)dx$
I don't have any idea.

REPLY [8 votes]: Integration term by term is valid because the convergence is uniform by the Weierstrass M-test, hence
$$\int_{\ln2}^{\ln3}f(x)dx=\int_{\ln2}^{\ln3}\sum_{n=1}^\infty n e^{-nx}dx=\sum_{n=1}^\infty n\int_{\ln2}^{\ln3} e^{-nx}dx=\sum_{n=1}^\infty2^{-n}-3^{-n}=\frac{1}{2-1}-\frac{1}{3-1}=\frac{1}{2}.$$