Here is the problem.
Let $k\in \mathbb{N}$. Suppose that there is a constant $C$ such that $|c_n|<\frac{C}{|n|^{k+1}}$ ($c_n$ here is the $n$th Fourier coefficient). Prove that $f(x)=\sum_{n\in\mathbb{N}}c_ne^{inx}$ has $k$ continuous derivatives.
I have absolutely no idea where to even start doing this. Any guidance would be greatly appreciated.