Computing the region of convergence for power series in many real variables: A ratio-like test

    Authors

    R. A. Van Gorder

    Comments

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    Abbreviated Journal Title

    Appl. Math. Comput.

    Keywords

    Multivariate power series; Convergence domain; Convergence test; Multivariate ratio test; ANALYTICAL REPRESENTATION; UNIFORM BRANCH; MULTIPLE SERIES; APPROXIMANTS; INTEGRALS; Mathematics, Applied

    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. (C) 2011 Elsevier Inc. All rights reserved.

    Journal Title

    Applied Mathematics and Computation

    Volume

    218

    Issue/Number

    5

    Publication Date

    1-1-2011

    Document Type

    Article

    Language

    English

    First Page

    2310

    Last Page

    2317

    WOS Identifier

    WOS:000294300800079

    ISSN

    0096-3003

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