Given a series:
$$f(x) = \sum_{n=1}^\infty (a_n \sin(nx)+b_n \cos(nx))$$
Provide a sufficient condition on the sequences {$a_n$} and {$b_n$} to be able to differentiate $f(x)$ term by term with
$$f'(x) = \sum_{n=1}^\infty (na_n \cos(nx)-nb_n \sin(nx))$$
I know we can interchange sums with differentiation if
(i)$\sum_{n=1}^\infty f_n(x_0)$ converges for some $x\in[a,b]$
(ii)$\sum_{n=1}^\infty f_n'$ converges uniformly to a differentiable function.
I can't seem to figure out how I go about it. My intuition is that $na_n$ and $nb_n$ should converge.
I also tried looking at it from Fourier series perspective but it didn't kead me to any conclusions. I would appreciate any help!