Let \(\sum_{\mu\in \mathbb{N}^n} a_\mu (x-x_0)^{\mu}\) be a multidimensional power series . Then
\begin{equation*} \mathcal{C}=\bigcup_{r>0} \{x\in \mathbb{R}^n\mid \sum_{\mu} \lvert a_\mu(y-x_0)^\mu\rvert< \infty, \quad \forall \lvert y-x\rvert< r\}. \end{equation*}For every point \(x\in \mathbb{C}\) there is some \(r>0\) such that the power series converge in \(B_r(x)\).
Remarks
- Since absolute convergence implies unconditional convergence the summands can be evaluated in an arbitrary order.
- Using Abel’s lemma one can show that \(\mathcal{C}=\mathring{\mathcal{B}}\), where \(y\in \mathcal{B}\) if all summands on \(y\) are bounded by some uniform constant.