How to tell if a convergent series of a sequence of functions is differentiable?

Question

Let's say I have $f_{n}:A=\mathbb{\left[-a,a\right]}\rightarrow\mathbb{R}$, where each $f_{n}$ is continuous on $\left[-a,a\right]$, with $a>0$. Also, I have that $f_{*}=\sum_{n=1}^{\infty}f_{n}$ converges uniformly on $A$ (does this matter for what I am asking?). How can I show that $f_{*}$ is differentiable on $B=\left(-a, a\right)$? Do I just need to show that $\sum_{n=1}^{\infty}f_{n}^{\prime}$ converges uniformly on $B$? If so, I don't see why this allows us to say that $f_{*}$ is differentiable on $B$, and an explanation would greatly be appreciated.

Answer 1

Suppose that $f_i'$ are continuous, and $f_i'\to g$ uniformly. Then I claim that $g=f'$ (where $f$ is your $f_*$). Now $$\int_{-a}^x g(t)\,dt=\lim_{i\to\infty}\int_{-a}^x f_i'(t)\,dt=\lim_{i\to\infty}(f_i(x)-f_i(-a)) =f(x)-f(a)$$ using uniform convergence of the $f_i'$. Now invoke the Fundamental Theorem of Calculus to get $g=f'$.