Open Access
2016 Completeness of Ordered Fields and a Trio of Classical Series Tests
Robert Kantrowitz, Michael M. Neumann
Abstr. Appl. Anal. 2016: 1-6 (2016). DOI: 10.1155/2016/6023273


This article explores the fate of the infinite series tests of Dirichlet, Dedekind, and Abel in the context of an arbitrary ordered field. It is shown that each of these three tests characterizes the Dedekind completeness of an Archimedean ordered field; specifically, none of the three is valid in any proper subfield of R. The argument hinges on a contractive-type property for sequences in Archimedean ordered fields that are bounded and strictly increasing. For an arbitrary ordered field, it turns out that each of the tests of Dirichlet and Dedekind is equivalent to the sequential completeness of the field.


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Robert Kantrowitz. Michael M. Neumann. "Completeness of Ordered Fields and a Trio of Classical Series Tests." Abstr. Appl. Anal. 2016 1 - 6, 2016.


Received: 12 April 2016; Revised: 25 September 2016; Accepted: 12 October 2016; Published: 2016
First available in Project Euclid: 17 December 2016

zbMATH: 06929379
MathSciNet: MR3574251
Digital Object Identifier: 10.1155/2016/6023273

Rights: Copyright © 2016 Hindawi

Vol.2016 • 2016
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