While reading the spectral radius formula (Theorem 18.9) in Real and Complex Analysis by Rudin, I found myself unable to understand the following: Rudin first defined a holomorphic function $f(z)$ on a ring region $\{z\mid |z|>r\ge 0\}$, then he proved that, when restricted to a smaller ring region $\{z\mid |z|>R\ge r\}$, $f(z)$ can be represented as a power series $\sum a_n z^n$, whereupon he concluded that not only on the smaller region, but also on the original bigger region where $f(z)$ is holomorphic, it still has the same power representation; in other words, $f(z)=\sum a_n z^n$ is still valid on $\{z\mid |z|>r\}$.
From my limited knowledge in complex analysis, I just can't recall any such properties of power series (or Laurent series, more accurately put) as the one stated above. I have a vague impression of analytic continuation (which I never mastered), is it relevant here?
Thanks in advance.