Lebesgue integrability of $\sum_{n=1}^\infty f_n(x)$, where $f_n(x):=ae^{-anx}-be^{-bnx}$

Question

Let $f_n(x):=ae^{-anx}-be^{-bnx}$ where $0<a<b$. Show that $\sum_{n=1}^\infty f_n(x)$ is Lebesgue integrable on $[0,\infty)$ and that $\int_0^\infty\sum_{n=1}^\infty f_n(x)=\ln(b/a)$.

Any hints on that? Lebesgue integrable means that $\int|f|<\infty$, but I don't know how to show that here and how to calculate the integral. I was thinking About using the Dominated Convergence Theorem, but couldn't find an estimate.