$\textbf{Q:}$ Is $\sum_{n\leq m} \exp(\delta\pi in)$ bounded for irrational $\delta\in R$?
The reason to ask this question is that $\log(1+z)$ has radius of convergence $1$ at $0$. And its expansion converges at the boundary of unit disk. It is trivial to see the convergence at finite order elements of $S^1\subset C$ via Abel summation. However, I do not see how to prove given any infinite order element $x$ of $S^1$, $\sum_{n\leq m} x^n$ has to be bounded, though it seems that the phase should eventually cancel out by dense argument for irrational number approximation.
$$\left|\sum_{n=1}^m e^{\delta \pi i n}\right| = \left| e^{\delta \pi i} \sum_{n=0}^{m-1} (e^{\delta \pi i})^n\right| = \left|e^{\delta \pi i} \frac{e^{\delta \pi i m}-1}{e^{\delta \pi i}-1}\right| \le \frac{2}{|e^{\delta \pi i}-1|}.$$