$\sum \limits_{k=1}^{\infty} \frac{1}{k} \cos 2\pi k \theta$ does not converge as $\theta \rightarrow 0?$

Question

We know that the series $H(\theta) := \sum \limits_{k=1}^{\infty} \frac{1}{k} \cos 2\pi k \theta$ is convergent for every $\theta \in (0,1)$ and for $\theta = 0$ the series tends to $+ \infty$. Is it true that the limit $\lim \limits_{\theta \rightarrow 0} H(\theta)$ does not exsist? How to prove that?

Answer 1

Knowing this series is convergent, you can find its sum using the formula $$ \sum_{k=1}^{\infty} \frac{z^k}{k} = -\log{(1-z)}. $$ Then the sum is $$ -\frac{1}{2}\left( \log{(1-e^{2\pi i \theta})}+\log{(1-e^{-2\pi i \theta})} \right), $$ and the limit of this as $\theta \to 0$ does not exist, since $\log{z}$ diverges as $z \to 0$.