Computing the region of convergence for power series in many real variables: A ratio-like test
Abbreviated Journal Title
Appl. Math. Comput.
Multivariate power series; Convergence domain; Convergence test; Multivariate ratio test; ANALYTICAL REPRESENTATION; UNIFORM BRANCH; MULTIPLE SERIES; APPROXIMANTS; INTEGRALS; Mathematics, Applied
We give an elementary proof that the region of convergence for a power series in many real variables is a star-convex domain but not, in general, a convex domain. In doing so, we deduce a natural higher-dimensional analog of the so-called ratio test from univariate power series. From the constructive proof of this result, we arrive at a method to approximate the region of convergence up to a desired accuracy. While most results in the literature are for rather specialized classes of multivariate power series, the method devised here is general. As far as applications are concerned, note that while theorems such as the Cauchy-Kowalevski theorem (and its generalizations to many variables) grant the existence of a region of convergence for a multivariate Taylor series to certain PDEs under appropriate restrictions, they do not give the actual region of convergence. The determination of the maximal region of convergence for such a series solution to a PDE is one application of our result. (C) 2011 Elsevier Inc. All rights reserved.
Applied Mathematics and Computation
"Computing the region of convergence for power series in many real variables: A ratio-like test" (2011). Faculty Bibliography 2010s. 2032.