How can one show that $\displaystyle\sum\limits_{n=1}^\infty\frac{1}{n^2}\text{cos}(nx) + \frac{1}{\sqrt{n}}\text{sin}(nx)$ can not be Fourier series of any $2\pi$-periodic continuous function?
The series $\displaystyle\sum\limits_{n=1}^\infty\frac{1}{n^2}\text{cos}(nx)$ is a Fourier series of its sum as it converges uniformly. So the problem appears when we add $\displaystyle\sum\limits_{n=1}^\infty\frac{1}{\sqrt{n}}\text{sin}(nx)$, but I have no ideas how to prove that an appropriate function does not exist.