This is Question 14 from Page 106 from the textbook Complex Variables: A Physical Approach, by Stephen G. Krantz.
Let $f$ be a holomorphic function defined on some open region $U \subseteq \mathbb C$ and fix a point $p \in U$. Prove that the power series expansion of $f$ about $p$ will converge absolutely and uniformly on any closed disc $D(p,r)$ with $r<\text{dist}(p,\partial U)$.
I know the following result: If $f$ is a holomorphic function on $U \subseteq \mathbb C$, with $p \in U$ and the closed disc $D(p,r) \subset U$, then $f(z)$ has a convergent power series in $D(p,r)$. Moreover, this power series is unique, and corresponds to the Taylor series at $z=p$.
The proof of this is given earlier in the textbook, on Page 101. I am unsure of how to proceed with the highlighted question above, in this case. Say the power series representation for $f$ is $$f(z)=\sum_{j=0}^\infty \frac{f^{(j)}(p)}{j!}(z-p)^j$$ Should I prove that this series is uniformly and absolutely convergent? If so, what would be the best way to go about doing this? I would greatly appreciate any help. I am not completely sure of what the question is asking, and how it is any different from the known result I have mentioned.
Hint: you'll want to use the M-test.