Convergence of a power series at a point

Question

Suppose a power series $\sum a_n z^n$ with $a_n,z \in \mathbb{C}$ converges for some single point $z_0$ in the complex plane.

Can we then say that it converges on any disk with radius $r\leq |z_0|$? I think yes since the power series has a radius of convergence, say $R$, where the series converges for $|z|<R$ and diverges for$ |z|>R$. So, since the series converges at a single point, R must be 'big enough' to include this point.

Am I correct in my thinking?

Answer 1

You can affirm that converges in a disk with $r<|z_0|$. Not equal. For example, $\sum \frac{1}{n}z^n$ converges for all $z\in \{z:|z|\leq 1\}-\{1\}$, but with $z=1$ diverges.

Then, if $z_0=i$, don't have convergency in the disk with $r=1$