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.

Remarks

Each power series converges at $z = c$, with sum $a_0$. The geometric series $\sum_{n \ge 0} z^n$ converges if and only if $|z| < 1$, and then its sum is $1/(1 - z)$. The set where the series converges contains an open disc and is contained in its closure, see disc of convergence. A formal power series is the same expression without any question of convergence.

Explanations

L1 human

A power series is a polynomial with infinitely many terms. For some values of $z$ the terms get small fast enough and the series has a sum, and for other values it does not. For example, $\sum z^n$ has a sum when $|z| < 1$, but not when $|z| \ge 1$, because then the terms do not go to $0$.