Let me introduce my problem: Let $C \subset \mathbb{C}$ be the unit circle of the complex plane and $Z=\left\{ z_n \right\}_{n \in \mathbb{N}} \subset C$ be a dense subset of the unit circle meaning that $\bar{Z}=C$ or otherwise put that for all $c \in C$ there is a subsequence $\left\{z_{n_{k}}\right\}_{k \in \mathbb{N}}$ of $Z$ such that $z_{n_{k}} \xrightarrow{k} c$. I want to know whether the following series converges:
\begin{align*} \sum_{j\in\mathbb{N}}z_n \end{align*}
For every $z_j$ on the unit circle there is it's anti-diametric $\hat{z_j}=-z_j$ and for all $\varepsilon>0$ we can find an index $\hat{j}\in\mathbb{N}$ so that $|\hat{z_j}-z_{\hat{j}}|<\varepsilon\Leftrightarrow |z_j+z_{\hat{j}}|<\varepsilon$.
If we group all members of the sequence this way (pairwise for a given $\varepsilon$) then the sum can be written in the following form:
\begin{align*} \sum_{j\in\mathbb{N}}z_n=\left(z_1+z_{\hat{1}}\right)+ \left(z_{k_2}+z_{\hat{k_2}}\right)+\ldots \end{align*}
Then by the triangular inequality:
\begin{align*} \sum_{j\in\mathbb{N}}z_n \le |\sum_{j\in\mathbb{N}}z_n| \le \left|z_1+z_{\hat{1}}\right|+ \left|z_{k_2}+z_{\hat{k_2}}\right|+\ldots \le \varepsilon + \varepsilon + \ldots \end{align*}
So this way I don't prove that the series converges. Is there some other way to prove it or it doesn't hold at all? (Anyway I think it was not a good idea to use the absolute value there...).
And a second question on that: If $\left\{a_n\right\}_{n\in \mathbb{N}}$ is a real sequence such that $\sum_{j\in\mathbb{N}}a_n$ converges then $\sum_{j\in\mathbb{N}}a_nz_n$ converges (following the same procedure described above). Can we somehow loosen the conditions on $\left\{a_n\right\}_{n\in \mathbb{N}}$ ? Are there weaker conditions on the sequence $a_n$ so that the series will converge?
Update 1: Does this series converge:
\begin{align*} \sum_{n\in\mathbb{N}}e^{2\pi\vartheta\alpha} \end{align*}
with $\alpha\in \mathbb{R} - \mathbb{Q}$ ?