On complex universal series (Q2848840)

A series \(\sum_n z_n\) of complex numbers is called universal if for every \(z \in \mathbb C\) there is a rearrangement \(\pi: \mathbb N \to \mathbb N\) such that \(z = \sum_{n=1}^\infty z_{\pi(n)}\). The main result of the paper says that if \(z \in \mathbb C\), \(z \notin \{1 , -1\}\) and \(|z| = 1\), then for every null sequence \((a_n)\) of nonnegative reals with \(\sum_{n=1}^\infty |a_n - a_{n+1}| < \infty\) and \(\sum_{n=1}^\infty |a_n| = \infty\) the corresponding series \(\sum_n a_n z^n\) is universal. In particular, the series \(\sum_n \frac{1}{n} z^n\) is universal.