Completeness of Ordered Fields and a Trio of Classical Series Tests
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 a...
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| Format: | Article |
| Language: | English |
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Wiley
2016-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2016/6023273 |
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| _version_ | 1832548228164222976 |
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| author | Robert Kantrowitz Michael M. Neumann |
| author_facet | Robert Kantrowitz Michael M. Neumann |
| author_sort | Robert Kantrowitz |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-db4d034455bf4e77ad479dc05849e589 |
| institution | Kabale University |
| issn | 1085-3375 1687-0409 |
| language | English |
| publishDate | 2016-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Abstract and Applied Analysis |
| spelling | doaj-art-db4d034455bf4e77ad479dc05849e5892025-02-03T06:41:57ZengWileyAbstract and Applied Analysis1085-33751687-04092016-01-01201610.1155/2016/60232736023273Completeness of Ordered Fields and a Trio of Classical Series TestsRobert Kantrowitz0Michael M. Neumann1Department of Mathematics, Hamilton College, 198 College Hill Road, Clinton, NY 13323, USADepartment of Mathematics and Statistics, Mississippi State University, Mississippi State, MS 39762, USAThis 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.http://dx.doi.org/10.1155/2016/6023273 |
| spellingShingle | Robert Kantrowitz Michael M. Neumann Completeness of Ordered Fields and a Trio of Classical Series Tests Abstract and Applied Analysis |
| title | Completeness of Ordered Fields and a Trio of Classical Series Tests |
| title_full | Completeness of Ordered Fields and a Trio of Classical Series Tests |
| title_fullStr | Completeness of Ordered Fields and a Trio of Classical Series Tests |
| title_full_unstemmed | Completeness of Ordered Fields and a Trio of Classical Series Tests |
| title_short | Completeness of Ordered Fields and a Trio of Classical Series Tests |
| title_sort | completeness of ordered fields and a trio of classical series tests |
| url | http://dx.doi.org/10.1155/2016/6023273 |
| work_keys_str_mv | AT robertkantrowitz completenessoforderedfieldsandatrioofclassicalseriestests AT michaelmneumann completenessoforderedfieldsandatrioofclassicalseriestests |