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

A complex sequence is a function $a\text{\hspace{0.17em}⁣}:\mathbb{N}\to \mathbb{C}$. 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 $S_0=0,S_{N+1}=S_N+a_N,S_N={\sum }_{n<N}^{\mathbb{C}}{a}_n\mathrm{.}$ 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 $\sum \mathrm{\mid }{a}_n\mathrm{\mid }$ of Series, partial sums, convergence and the sum, divergence, and the tail series converges. For a bijection $\sigma \text{\hspace{0.17em}⁣}:\mathbb{N}\to \mathbb{N}$, 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 ${\sum }_{n=0}^{\mathrm{\infty }}{c}_n(z-a{)}^n,$ 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 $Q(x):={\sum }_{n=0}^{\mathrm{\infty }}\mathrm{\mid }{c}_n\mathrm{\mid }{x}^n$ 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 $[0,+\mathrm{\infty }]$. Thus the coefficient sequence, real variable, and center of the comparison series are all explicit.