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Completeness of Ordered Fields and a Trio of Classical Series Tests

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  • Robert Kantrowitz
  • Michael M. Neumann

Abstract

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 . 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.

Suggested Citation

  • Robert Kantrowitz & Michael M. Neumann, 2016. "Completeness of Ordered Fields and a Trio of Classical Series Tests," Abstract and Applied Analysis, Hindawi, vol. 2016, pages 1-6, November.
    • Handle: RePEc:hin:jnlaaa:6023273
    DOI: 10.1155/2016/6023273
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