\begin{equation*} \sum_{n=0}^{\infty} a_n(z-z_0)^n \end{equation*}

has the radius of convergence \(r>0\) if and only if for every \(00\) such that for all \(n\in \mathbb{N}\)

\begin{equation*} \lvert a_n\rvert\le \frac{C}{R^n}. \end{equation*}

[1, Corollary 1.10]

For multidimensional power series there is a similar result. Let \(\sum_{\mu} a_\mu (x-x_0)^\mu\) be a multidimensional power series and \(\mathcal{C}\) its domain of convergence . For every \(y\in \mathcal{C}\) there are constants \(C>0\) and \(\varepsilon>0\) such that

\begin{equation*} \lvert a_\mu\rvert \le \frac{C}{\prod_{j=1}^{n} \lvert y_j-(x_0)_j\rvert^{\mu_j}}. \end{equation*}

[1, Lemma 2.1.10]

Proof

This is a consequence of the Cauchy-Hadamard theorem

References Link to heading

S. Krantz and H. Parks, A Primer of Real Analytic Functions. Boston, MA: Birkhäuser Boston, 2002. doi:10.1007/978-0-8176-8134-0