Convergence of power series at boundary

Question

Suppose that $f(z) = \sum a_n z^n$ for $z$ in the unit disc. If $\sum a_n (z_0)^n$ diverges for some $z_0$ on the unit circle, does this necessarily imply that $\lim_{z \rightarrow z_0} \sum a_n z^n$ does not exist?

EDIT: What about if we know that $\sum a_n (z_0)^n$ is in fact, infinite?

Answer 1

No. Consider the function $\frac{1}{1+ z^2} = \sum_{n=0}^{\infty} (-1)^n z^{2n}$ in $\{|z| < 1\}$ and $z_0 = 1$.