Application of Differentiation using Lebesgue Dominated Convergence Theorem

Question

I am working through old Measure Theory homework assignments and this question is giving me some trouble. It states:

Let {$f_n$} be a sequence of continuous and differentiable functions on the interval $[a,b]$. Assume that for each $t$ in $[a,b]$, $\sum_{n}|f_n(t)|$ is convergent and $|f^{'}_n(t)|\le g(n)$ where $\sum_{n}g(n)$ is convergent. Show that the function $F(t)=\sum_{n}f_n(t)$ is differentiable on $(a,b)$ and $F'(t)=\sum_{n}f^{'}_{n}(t)$.

I know this is an application of the Lebesgue Dominated Convergence Theorem but cannot figure out how to set it up. Any hints, helps, or leads would be nice! Thanks.

Answer 1

I'd start by writing $$ f_n(t) =f_n(a)+\int_a^t f_n'(s)\,ds $$ for $t\in[a,b]$ and $n\ge 1$. Now sum with respect to $n$ on both sides. The Weierstrass $M$-test should come in handy to show that $\sum_n f'_n(s)$ converges uniformly and that the sum, call it $h(s)$, is a continuous function of $s$.