continuity of series of powers of cosine

Question

I'm trying to understand the continuity of $$\sum_{n=1}^{\infty}\frac{\cos^n(nx)}{n^{3/2}}.$$ I believe these series converge and it can be proved by comparison test. I also figured that $$\lim_{n \rightarrow \infty} \cos^n(nx)$$ is discontinuous at $x=\frac{\pi k}{n}$ where $k$ is an integer. My intuition tells me that the series should be discontinuous at the same $x$. However, I don't know how to prove or disaprove this.