Taylor series of the exponential function

Theorem: Let $z_0 \in \mathbb{C}$. The following Taylor series holds for all $z \in \mathbb{C}$: $$e^z = \displaystyle\sum_{k=0}^{\infty} \dfrac{(z-z_0)^k}{k!},$$ where $e^z$ is the exponential function.

Proof: █