Decay of Fourier Coefficients and Smoothness
Hint: Use the theorem
If $g_n \to g$ pointwise, and $g_n$ differentiable, and that $g'_n \to h$ uniformly, then $g$ is differentiable with derivative $h$.
Let $g_n$ in this case be the Fourier partial sums $\sum_{|k| \leq n} a_k e^{ikx}$.