Power Series Are Holomorphic and Are Differentiated Term by Term

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; convergence and sums of series in $\mathbb{C}$ are as in Series of Complex Numbers: Partial Sums, Convergence, the Sum and Absolute Convergence (ii) and absolute convergence as in Series of Complex Numbers: Partial Sums, Convergence, the Sum and Absolute Convergence (iii); complex derivatives are as in The Complex Derivative at a Point (ii), and holomorphic functions and their derivatives as in Holomorphic Functions and Their Derivatives (i) and Holomorphic Functions and Their Derivatives (ii).

(i) Power series on a disk. Let $R>0$ be a real number such that $\sum_{k=1}^{\infty}|c_{k}|\,\rho^{k-1}$ converges for every real $\rho$ with $0<\rho<R$, and let $D=\{z\in\mathbb{C}:|z|<R\}$. Then $D$ is open in $(\mathbb{C},d_{\mathbb{C}})$; for every $z\in D$ the series

$$\sum_{k=1}^{\infty}c_{k}z^{k-1}\qquad\text{and}\qquad\sum_{k=1}^{\infty}k\,c_{k+1}z^{k-1}$$

converge absolutely; and the function $f:D\to\mathbb{C}$, $f(z)=\sum_{k=1}^{\infty}c_{k}z^{k-1}$ (Complex Power Series: the Sequence of Terms and the Sum as a Function on the Set of Convergence (iii)), is holomorphic on $D$, with

$$f'(z)=\sum_{k=1}^{\infty}k\,c_{k+1}z^{k-1}\qquad(z\in D).$$

(ii) Power series on the plane. Suppose that $\sum_{k=1}^{\infty}|c_{k}|\,\rho^{k-1}$ converges for every real $\rho>0$. Then for every $z\in\mathbb{C}$ the two series displayed in (i) converge absolutely; and the function $f:\mathbb{C}\to\mathbb{C}$, $f(z)=\sum_{k=1}^{\infty}c_{k}z^{k-1}$, is holomorphic on $\mathbb{C}$, with

$$f'(z)=\sum_{k=1}^{\infty}k\,c_{k+1}z^{k-1}\qquad(z\in\mathbb{C}).$$