Where is the series $\displaystyle\sum_{n=1}^{\infty}\dfrac{\cos(nx)}{n}$ pointwise convergent? Where is the series $\displaystyle\sum_{n=1}^{\infty}\dfrac{\cos(nx)}{n}$ pointwise convergent?
I tried to apply the Dirichlet's test but I couldn't.

Where did you find trouble in applying it? Hint Find a closed form for $$\sum_{k=0}^n\cos kx$$ that let's you decided when this sum is bounded.  A good idea might be to note this is $$\Re\left(\sum_{k=0}^n e^{ikx}\right)$$ 
