Give an example, if possible, of a complex power series centered at $0$ and convergent for all complex $z$, with $Re(z)=8$, but divergent for all other complex $z$.
The way I thought of this was: Clearly we can find complex $z_1$, with $Re(z_1)=8$, but $|z_1|>R$, for any given positive $R$, so that the series converges for that $z_1$. Then, by Abel's Theorem, the series has to converge for all complex $z_2$, with $|z_2|<|z_1|$, regardless of $Re(z_2)$. So a power series with these properties does not exist. Is my argument correct?
It doesn't exist. Since it converges at $8$, it converges at every $z$ such that $\lvert z\rvert<8$.