Steps to determine the interval of continuity of $f(x)= \sum_{n=2}^{\infty} \frac{(\sin{nx})^2}{\sqrt{n}}$

Question

I am trying I determine the interval of continuity of $$f_n(x)= \sum_{n=2}^{\infty} \frac{(\sin{nx})^2}{\sqrt{n}}$$ I tried to find the domain by Dirichlet's test, but the sum $\sum _{n=1}^{\infty }(\sin (nx)^2)$ does not converge. I also tried the ratio test on which I got stuck on. The root test was inconclusive. What other tests can I apply?

Answer 1

$$\sum_{n=1}^\infty \frac{\sin^2(nx)}{\sqrt{n}} = \sum_{n=1}^\infty \frac{1 - \cos(2nx)}{2\sqrt{n}}$$

Of course, $\sum_n 1/\sqrt{n}$ diverges, while $\sum_n \cos(2nx)/\sqrt{n}$ converges by Dirichlet's test whenever $\sin(x) \ne 0$. Therefore your series must diverge whenever $\sin(x) \ne 0$.