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- W2546531566 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 <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M1><mml:mrow><mml:mi mathvariant=double-struck>R</mml:mi></mml:mrow></mml:math>. 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." @default.
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- W2546531566 date "2016-01-01" @default.
- W2546531566 modified "2023-10-13" @default.
- W2546531566 title "Completeness of Ordered Fields and a Trio of Classical Series Tests" @default.
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- W2546531566 doi "https://doi.org/10.1155/2016/6023273" @default.
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