Where is the series $\displaystyle\sum_{n=1}^{\infty}\dfrac{\cos(nx)}{n}$ pointwise convergent?

Question

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.

Answer 1

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 $$\Re\left(\sum_{k=0}^n e^{ikx}\right)$$

Answer 2

Hint

Following Pedro Tamaroff, may be, you could consider two summations $$I=\displaystyle\sum_{n=1}^{\infty}\dfrac{\cos(nx)}{n}$$ $$J=\displaystyle\sum_{n=1}^{\infty}\dfrac{\sin (nx)}{n}$$ Then $$I+iJ=\displaystyle\sum_{n=1}^{\infty}\dfrac{e^{inx}}{n}=\displaystyle\sum_{n=1}^{\infty}\dfrac{y^{n}}{n}=-\log (1-y)$$ where $y=e^{ix}$ and use the real part of the complex logarithm.