Complex series, absolute convergence, complex power series, and radius of convergence

A complex sequence is a function $a\colon\mathbb N\to\mathbb C$. Since the additive reduct $(\mathbb C,+,0)$ of the complex field is a commutative monoid, the finite-list construction of The product $g_0 g_1 \cdots g_{n-1}$ of a finite list in a monoid, by recursion, with the empty product ($n = 0$) equal to the identity, read additively, defines its complex partial sums by

$$S_0=0,\qquad S_{N+1}=S_N+a_N,\qquad S_N=\sum_{n<N}^{\mathbb C}a_n.$$

The superscript is omitted when the summands already make the codomain clear.

The complex series $\sum a_n$ converges to $s\in\mathbb C$ when $S_N\to s$ in the metric of The Euclidean metric, convergence, Cauchy sequences, and continuity on the complex plane; then $s$ is its sum. This is the same finite sum and convergence as the vector-series definition on $\mathbb R^2$ in Series of vectors in $\mathbb{R}^n$, absolute convergence, rearrangement, and the set of rearrangement sums. The series converges absolutely when the real series $\sum |a_n|$ of Series, partial sums, convergence and the sum, divergence, and the tail series converges. For a bijection $\sigma\colon\mathbb N\to\mathbb N$, its rearrangement along $\sigma$ is the complex series $\sum a_{\sigma(n)}$.

Let $(c_n)$ be a complex sequence and $a\in\mathbb C$. The complex power series centered at $a$ with coefficients $(c_n)$ is

$$\sum_{n=0}^{\infty}c_n(z-a)^n,$$

with powers from Integer powers in the complex field and the preceding complex partial sums. To define its radius without ambiguity, form the specific real power series

$$Q(x):=\sum_{n=0}^{\infty}|c_n|x^n$$

centered at $0$. The radius of convergence of the complex power series is, by definition, the radius of this $Q$ in A real power series about a centre, its interval of convergence, and its radius in $[0,+\infty]$. Thus the coefficient sequence, real variable, and center of the comparison series are all explicit.