 Solution

Repeated integration by parts in the cosine-coefficient integral shows $|c_k|\le C_r{k}^{-r}$ for every $r$ when $f$ is smooth (endpoint terms vanish after using the even periodic extension). Parseval then bounds the squared tail by $C_r{\sum }_{k>N}{k}^{-2r}$, which is $O(N^{-2r+1})$. Choosing $r$ arbitrarily large proves spectral convergence.