How can I prove that $$\sum_{n=1}^\infty \frac{\sin(nx)}{\sqrt{n}}$$ converges pointwise on $[-\pi, \pi]$ but not uniformly?
For the pointwise part, I tried to prove it by comparison, using $$\sum_{n=1}^\infty \frac{\sin(nx)}{\sqrt{n}} \leq \sum_{n=1}^\infty \frac{nx}{\sqrt{n}} = \sum_{n=1}^\infty x\sqrt{n},$$ which does not converge.
I also tried $$ \sum_{n=1}^\infty \left| \frac{\sin(nx)}{\sqrt{n}}\right| \leq \sum_{n=1}^\infty \left| \frac{1}{\sqrt{n}}\right|,$$ which does not converge either.