How to show that $$ \sum_{n=2}^\infty \frac{\sin{(nx)}}{\log n} $$ not the Fourier series of any function?
I have shown that the series is convergent by Dirichlet test.
Let $a(n)=\frac{1}{\log n}$. What is $\sum (a(n))^2$, to apply Parseval's theorem?
Suppose that it is the Fourier series of $f(x)$ in $(-\pi,\pi)$, namely, $$ f(x)=\sum_{n=2}^\infty \frac{\sin(nx)}{\log n},\quad x\in(-\pi,\pi).$$ Then by Pareraval's Identity, $$\int_{-\pi}^\pi f(x) \, dx=\sum_{n=2}^\infty\frac{1}{\log^2n}=\infty.$$