Abel's Theorem (complex version). Let $G(z)=\sum_{n=0}^\infty c_nz^n$ and let the series of coefficients $\sum_{n=0}^\infty c_n$ converge. Then $$ \lim_{z\to 1} G(z)=G(1), $$ 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 the associated wikipedia entry for a sketch of the proof. Also there, a counterexample is given as follows: the series $$ \sum_{n=1}^\infty \frac{z^{3^n}-z^{2\cdot3^n}}{n} $$ 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 see an example of power series for which 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 a circumference inside the unit circle $\{z : |z|<1\}$ which is tangential to the unit circle at $z=1.$
Thanks in advance!
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.