Convergence of power series at the boundary

Question

Consider the following complex power series $$ \sum_{n\geq 1}{\frac{ni^n}{2^n}{z^{n-1}}} $$ By the root test, I have concluded that the disc of convergence is $D:=D(0,2)$. Then, I would like to study the convergence of the series in $\partial D$. Considering $z=2e^{i\theta}\in \partial D$, $\theta \in \mathbb{R}$, the series becomes $$ \frac{1}{2}\sum_{n\geq 1}{n i^n e^{i(n-1)\theta}} $$ which is not absolutely convergent and, indeed, $$ \lim_{n} n i^n e^{i(n-1)\theta} \neq 0 $$ for every $\theta \in \mathbb{R}$, so the series cannot be convergent at the boundary. Is this reasoning true, or am I missing some aspect of the boundary convergence of complex power series? Thanks in advance for your answers.

Answer 1

You are correct. To be more precise: let $a_n:=n i^n e^{i(n-1)\theta}$, then $|a_n|=n$ for all $n$, thus $(a_n)$ does not converge to $0.$