Counter example for Stolz sector in complex version of Abel's Theorem

Question

(Abel's Theorem: complex version) Let $G(z)=\sum_{n=0}^\infty c_nz^n$ and $\sum_{n=0}^\infty c_n$ converges. Then $\displaystyle\lim_{z\to 1} G(z)=G(1),$ provided that $z \to 1$ within a Stolz sector (non-tangential approach region), that is, a region of the open unit disk where $|1-z| \le M(1-|z|)$ for some $M>0.$ See https://en.wikipedia.org/wiki/Abel%27s_theorem. Here, a counterexample is given as follows: $\sum_{n=1}^\infty \frac{z^{3^n}-z^{2\cdot3^n}}{n}$ converges to $0$ at $z=1,$ but it diverges at $z_n=exp(i \pi/3^n)$. However, it is not a power series.

I'd like to know an example that Abel's Theorem does not hold if $z_n \to 1$ and $z_n$ is not in a Stolz sector, i.e., $z_n$ is on the curve in a unit circle $\{z : |z|<1\}$ which is tangential to the unit circle at $z=1.$

Thanks in advance!

Answer 1

The counterexample provided in the OP is INDEED a power series $\sum c_nz^n$ with: $$ c_n=\left\{\begin{array}{rll} \frac{1}{k} & \text{if} & n=3^k, \\ -\frac{1}{k} & \text{if} & n=2\cdot 3^k, \\ 0 & \text{otherwise} \end{array} \right. $$ which provides shows that Abel's Theorem does not hold for tangential limits.