Title
Computing The Region Of Convergence For Power Series In Many Real Variables: A Ratio-Like Test
Keywords
Convergence domain; Convergence test; Multivariate power series; Multivariate ratio test
Abstract
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. © 2011 Elsevier Inc. All rights reserved.
Publication Date
11-1-2011
Publication Title
Applied Mathematics and Computation
Volume
218
Issue
5
Number of Pages
2310-2317
Document Type
Article
Personal Identifier
scopus
DOI Link
https://doi.org/10.1016/j.amc.2011.07.052
Copyright Status
Unknown
Socpus ID
80052262702 (Scopus)
Source API URL
https://api.elsevier.com/content/abstract/scopus_id/80052262702
STARS Citation
Van Gorder, Robert A., "Computing The Region Of Convergence For Power Series In Many Real Variables: A Ratio-Like Test" (2011). Scopus Export 2010-2014. 1888.
https://stars.library.ucf.edu/scopus2010/1888