Analyticity and Complex Differentiability

Question

I'm seeing two different definitions for analytic functions.

1. A function $\Phi: \mathcal{D}\rightarrow \mathbb C$ is analytic if, for every $z_0 \in \mathcal{D}\subseteq \mathbb C$, there exists a power series converging to $\Phi(z)$ on some neighborhood of $z_0$.
2. A function $\Phi: \mathcal{D}\rightarrow \mathbb C$ is analytic if $\Phi$ is complex differentiable on an open set $\mathcal{D}$.

I get that power series functions are differentiable, but I don't see how these two definitions say the same thing. Is there any explanation for that?