given $\displaystyle{ z \in \mathbb{C}, |z|=1 , z\ne 1 ; \sum_{n\in \mathbb{N}} \frac{z^{n}}{n}}$ I will show that:
i) the partial sum of this series is bounded. , ii) that $\displaystyle{\sum_{n\in \mathbb{N}}\frac{z^{n}}{n}} converges$, iii) $\displaystyle{\sum_{n\in \mathbb{N}}\frac{sin(an)}{n} }$ converges.
I know this is a power series...
i) How can one show the partial sum is bounded?
ii) $\displaystyle{|\sum \frac{z^{n}}{n}|} \le \sum|\frac{z^{n}}{n}| $
iii) write in polar form cos+isin then it follows from ii)
Does anybody see a way? Please do tell.