Prove that the function $f(x) = \sum_{k =1}^{\infty} \frac{\sin(kx)}{2^{k}}$ is infinitely differentiable

Question

Prove that the function $f(x) = \sum_{k =1}^{\infty} \frac{\sin(kx)}{2^{k}}$ is infinitely differentiable

This is a practice problem for an exam I have coming soon. I am trying to study, but I cannot figure out how to do this problem. I think that the correct approach would be to show that the function $f$ is analytic, which implies that $f$ is infinitely differentiable.

I'm not sure if this is the correct approach, and how to do it, though.

Any help is appreciated.