Is the series $\sum_{n=1}^{\infty} \frac{\sin nx}{n}$ is uniformly convergent on $[0,1]$?

Question

Is the series $\sum_{n=1}^{\infty} \frac{\sin nx}{n}$ uniformly convergent on $[0,1]$?

My work : since $|\sin nx|\le 1$ so $\sum_{n=1}^{\infty} \frac{\sin nx}{n} \le \sum_{n=1}^{\infty} \frac{1}{n}$ which is divergent

so my answer is no, it's not uniformly convergent on $[0,1]$.

Please verify

Thank you

Answer 1

This requires some knowledge of the theory of Fourier series., It is known that if $a_n$ decreases to $0$ then $\sum a_n sin (nx)$ is uniformly convergent iff $na_n \to 0$. In this case $na_n=1$ so the series does not converge uniformly. Ref. Fourier Series: A Modern Introduction by Edwards. [See section 7.2].