Complex Power Series: the Sequence of Terms and the Sum as a Function on the Set of Convergence

In the setting of Complex Analysis: Standing Notation, let $(c_{k})$ be a sequence in $\mathbb{C}$. For $k\in\mathbb{N}$, $k-1\in\mathbb{N}_{0}$ is the difference; the factorial $n!$ of $n\in\mathbb{N}_{0}$ is read in $\mathbb{C}$; and convergence of a series in $\mathbb{C}$ and its sum are as in Series of Complex Numbers: Partial Sums, Convergence, the Sum and Absolute Convergence (ii).

(i) For every $u\in\mathbb{C}$ there is exactly one sequence in $\mathbb{C}$ whose $k$-th term is $c_{k}u^{k-1}$ for every $k\in\mathbb{N}$.

(ii) For every $n\in\mathbb{N}_{0}$, $n!\neq0$ in $\mathbb{C}$; and there is exactly one sequence in $\mathbb{C}$ whose $k$-th term is $1/(k-1)!$ for every $k\in\mathbb{N}$.

(iii) Let $D\subseteq\mathbb{C}$ be such that, for every $u\in D$, the series $\sum_{k=1}^{\infty}c_{k}u^{k-1}$ of the sequence of (i) converges. Then there is exactly one function $h:D\to\mathbb{C}$ with $h(u)=\sum_{k=1}^{\infty}c_{k}u^{k-1}$ for every $u\in D$.