# Prove that the sum of a series is differentiable

Prove that the series

$$\sum_{k=2}^∞ \sin (kx)/k\ln^2(k)$$

is absolutely and uniformly convergent on $$\mathbb{R}$$. If the sum of the series is denoted by $$f(x)$$ prove that $$f$$ is differentiable at each point of ($$0; 2\pi$$), and find $$f'$$.

I already proved that it absolutely and uniformly converges, but I don't know how to prove that it is differentiable. Any help please?

• Integration term-by-term is easier to justify than differentiation. So consider the term-by-term derivative of your series, and investigate whether it can be integrated. – GEdgar May 9 at 12:58