For this I use the root criterion $\lim\limits_{n\to \infty}\sqrt[n]{|a_n|}$.
How the series is $\sum_{n=1}^{\infty}z^ne^{in}$ we can see that $a_n=e^{in}$. Then
$\lim\limits_{n\to \infty}\sqrt[n]{|a_n|}=\lim\limits_{n\to \infty}\sqrt[n]{|e^{in}|}=\lim\limits_{n\to \infty}\sqrt[n]{(e^i)^n}=\lim\limits_{n\to \infty}e^i=e^i$.
Then $R=e^i$
But they give me the following conditions: Let $r$ be the radius of convergence, then
if $R\in \mathbb{R}$ whit $R>0$ then $r=\frac{1}{R}$
if $R=0$ then $r=\infty$
inf $R=\infty$ then $r=0$
But I found $R=e^i$ and it does not meet any of the three conditions. What can be done in these cases?