I know the following theorems but don’t know their usefulness.
If a series $\{f_n\}$ of Riemann integrable functions on $[a, b]$ uniformly converges to $f$, $f$ is Riemann integrable and $\lim\limits_{n\to\infty}\int_a^b f_n=\int_a^b f$.
If a series $\{f_n\}$ of $C^1$ functions on $[a, b]$ converges at one point and $\{f_n’\}$ converges uniformly, it uniformly converges to a $C^1$ function $f$ and $\lim\limits_{n\to\infty}f_n’=f’$.
What can these be used for?
I know that power series can be differentiated/integrated term by term in the circle of convergence, but I don’t use the above theorems to prove that.
