I am having a bit of trouble trying to show that $\sum_{n = 1}^{\infty} \frac{\sin(nx)}{n^{\alpha}}$ for $\alpha \in \left(0, \frac{1}{2}\right)$ is not the Fourier series of a RI-function.
I have shown that the series is pointwise convergent by Dirichlet's test for a fixed $x$, since the sequence of partial sums of $\sin(nx)$ is bounded, i.e. \begin{equation*} \left|\sum_{k = 1}^{n} \sin(kx)\right| \leq \frac{1}{\sin\left(\frac{x}{2}\right)} \end{equation*} and that the sequence $\frac{1}{n^{\alpha}} \to 0$ and is a decreasing sequence. So it should converge pointwise. But I am unsure how to proceed to show that the series cannot be the Fourier series of a RI-function. A hint that I was given was to use either Bessel's Inequality or Parseval's Identity. Some advice would be helpful.