Power series in the complex plane

Definition

Let $c \in \mathbb{C}$ and let $(a_n)_{n \ge 0}$ be a sequence of complex numbers. The power series with center $c$ and coefficients $a_n$ is the series $$\sum_{n=0}^{\infty} a_n (z - c)^n$$ in the complex variable $z$. The series converges at a point $z \in \mathbb{C}$ if the partial sums $\sum_{n=0}^{N} a_n (z - c)^n$ have a limit in $\mathbb{C}$ as $N \to \infty$. The sum of the series is the function that maps each such $z$ to this limit.