Let \(f\) be a power series . Then it is analytic on its domain of convergence. [1, Corollary 1.2.4]

The same is true for multidimensional power series on there domain of convergence [1, Proposition 2.2.7].

References Link to heading

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