A Pringsheim-type convergence criterion for continued fractions in Banach algebras

We consider continued fractions in Banach algebras, that is b(0) + a(1)(b(1)+a(2)(b(2) + ...)(-1))(-1), where (b(n))(n is an element of N0) and (a(n))(n is an element of N) are sequences of elements of some Banach algebra. We prove that parallel to b(n)(-1)parallel to + parallel to a(n)b(n)(-1)parallel to <= 1 for n = 1, 2, ... is a sufficient condition for convergence. This result is an exact generalization of the Sleszynski-Pringsheim convergence criterion for complex continued fractions, and improves on all known results. (C) 2012 Elsevier Inc. All rights reserved.