I have calculated the Fourier series as given in the question to be
$$f(x)=\frac{8}{15}+\frac{48}{\pi^4}\sum_{1}^{\infty }(-1)^{n+1}\frac{\cos(n\pi x)}{n^4}$$ I am asked to comment on the differentiability and the order of its terms in the Fourier series as $n\to\infty$ Is the following argument correct, the periodic extension of $f''(x)$ belongs to $C(\mathbb{R})$ and $f''(x)$ is continuously differentiable but the periodic extension of $f'''(x)$ is not in $C(\mathbb{R})$ so the fourth derivative will not converge to its Fourier series when differentiated term wise. So it is 3 times differentiable and the terms are $O(1/n^4)$.