Background
I was wondering whether there any books, articles, or other in-depth treatments of infinite series involving the roots of unity. Let $\omega_{n} := e^{2 \pi i / n}$ be the $n$'th root of unity. Even though some (finite) sums involving them are known (see e.g. this question), I haven't found any resources on for example:
$$S_{1} := \sum_{n=1}^{\infty} \frac{\omega_{n}}{n!} \tag{1}\label{1} \quad, $$ or the rational zeta series $$ S_{2} := \sum_{k=2}^{\infty} \left( \zeta(k)-1 \right) \omega_{k} \quad. \tag{2}\label{2} $$
As far as I know, there aren't any particular applications for these kinds of series - this question arose mainly out of curiosity.
Question
Is there any literature on the evaluation of series like equation \eqref{1}, equation \eqref{2}, or other series involving the $n$'th roots of unity?
