In our analysis class today, our teacher wanted us to prove the following theorem, or according to him, known as Abel's theorem:
If $\sum\limits_{n=0}^{\infty} a_n (z-z_0)^n$ with $a_n \in \mathbb{C}$ converges at $z_1 \ne z_0$, then the power series converges for all $z$, such that $|z-z_0| < |z_1 - z_0|$.
I have a hard time understanding what this theorem is trying to say. Where did $z_1$ come from? And how will one be able to proof? I tried looking for this theorem online but I couldn't find anything.
Any help will be greatly appreciated. Thanks in advance!