If I have a holomorphic function on the unit disc, do I know anything about the radius of convergence of its series expansion about zero? I'm looking at a proof that assumes only that
$f : \mathbb{D} \rightarrow \mathbb{D}$ is holomorphic with $f(0) = 0$
The first step in the proof is to "expand $f$ in a power series centered at $0$ and convergent in all of $\mathbb{D}$."  
In other words, $f(z) = a_0 + a_1 z + a_2 z^2 + ...$ inside the disc. Is this a valid step? How do I know that the function's series expansion is valid in the whole disc?

Following Tunococ's link, I found the following: "the fact that the radius of convergence is always the distance from the center a to the nearest singularity; if there are no singularities (i.e., if ƒ is an entire function), then the radius of convergence is infinite. Strictly speaking, this is not a corollary of the theorem but rather a by-product of the proof."
Since there are no singularities, then the series is valid for the whole disc!
