Question

I wonder whether the sequence $s_{n} = \sum_{k = 1}^{n} \sin k$ is bounded.

The answer seems no, but I have no idea how to prove this from the irrationality of $\pi$.

Answer 1

In this particular case, use $s_n(\theta)=\sum_{k=0}^n\sin(k\theta)=\Im(\sum_{k=0}^n\exp(ki\theta))=\dots=\frac{\sin(n\theta/2)\sin((n+1)\theta/2)}{\sin (\theta/2)}$