Science Journal of Applied Mathematics and Statistics

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Improvement of the Raabe-Duhamel Convergence Criterion Generalized

Received: 18 July 2023    Accepted: 23 August 2023    Published: 16 November 2023
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Abstract

In Mathematics, convergence tests are methods of testing for the convergence, conditional convergence, absolute convergence, interval of convergence or divergence of an infinite series. There are many criterion for testing the convergence of an infinite series: Cauchy, D’Alembert, Riemann, Bertrand and so one. One of the most important is the Raabe-Duhamel convergence criterion which asserts that: Given an infinite series nun with positive terms un and assuming that the following expansion holds , as n → ∞. Then the series nun converges if λ > 1 and diverges if λ < 1. However no conclusion can be made if λ = 1. Indeed the infinite series and satisfy both the expansion with λ = 1. The first one converges and the second one diverges. The aim of the present paper deals with the convergence of a generalized Riemann-Bertrand infinite series. This will allows us to improve the expansion so that something can be said if l = 1: this corresponds to the improvement of the Raabe-Duhamel convergence criterion. This improvement is based on the convergence of a new type of infinite series. These type of series are generalization of the Riemann and Bertrand infinite series.

DOI 10.11648/j.sjams.20231103.11
Published in Science Journal of Applied Mathematics and Statistics (Volume 11, Issue 3, June 2023)
Page(s) 44-47
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This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Infinite Series, Raabe-Duhamel Convergence Criterion, Riemann and Bertrand Infinite Series

References
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[3] S. Balac and F. Sturm, Algèbre et analyse: cours de mathématiques de première année avec exercices corrigés, PPUR presses polytechniques, 2003.
[4] B. V. Belorusets, Finding the domain of convergence of volterra series, Autornation and Remote Control, 51: 1-6, 1990.
[5] N. Boboc, Mathematical Analysis (in Romanian), vol. 1, Editura Universitatii Bucuresti, 1999.
[6] Bonar, Daniel D., and Michael J. Khoury Jr, Real infinite series, Vol. 56. American Mathematical Soc., 2018.
[7] I. Bucur, On a generalized gauss convergence criterion, Rom. J. Math. Comput. Sci. 5 (1) (2015), 76-83.
[8] J. A. Guthrie and J. E. Nymann, The topological structure of the set of subsums of an infinite series, Colloquium Mathematicum. Vol. 2. No. 55. 1988.
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[10] Hirschman, Isidore Isaac, Infinite series, Courier Corporation, 2014.
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  • APA Style

    Hadji Abdoulaye Thiam, E., Moussa Niang, P. (2023). Improvement of the Raabe-Duhamel Convergence Criterion Generalized. Science Journal of Applied Mathematics and Statistics, 11(3), 44-47. https://doi.org/10.11648/j.sjams.20231103.11

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    ACS Style

    Hadji Abdoulaye Thiam, E.; Moussa Niang, P. Improvement of the Raabe-Duhamel Convergence Criterion Generalized. Sci. J. Appl. Math. Stat. 2023, 11(3), 44-47. doi: 10.11648/j.sjams.20231103.11

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    AMA Style

    Hadji Abdoulaye Thiam E, Moussa Niang P. Improvement of the Raabe-Duhamel Convergence Criterion Generalized. Sci J Appl Math Stat. 2023;11(3):44-47. doi: 10.11648/j.sjams.20231103.11

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  • @article{10.11648/j.sjams.20231103.11,
      author = {El Hadji Abdoulaye Thiam and Papa Moussa Niang},
      title = {Improvement of the Raabe-Duhamel Convergence Criterion Generalized},
      journal = {Science Journal of Applied Mathematics and Statistics},
      volume = {11},
      number = {3},
      pages = {44-47},
      doi = {10.11648/j.sjams.20231103.11},
      url = {https://doi.org/10.11648/j.sjams.20231103.11},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.sjams.20231103.11},
      abstract = {In Mathematics, convergence tests are methods of testing for the convergence, conditional convergence, absolute convergence, interval of convergence or divergence of an infinite series. There are many criterion for testing the convergence of an infinite series: Cauchy, D’Alembert, Riemann, Bertrand and so one. One of the most important is the Raabe-Duhamel convergence criterion which asserts that: Given an infinite series ∑nun with positive terms un and assuming that the following expansion holds , as n → ∞. Then the series ∑nun converges if λ > 1 and diverges if λ  and  satisfy both the expansion with λ = 1. The first one converges and the second one diverges. The aim of the present paper deals with the convergence of a generalized Riemann-Bertrand infinite series. This will allows us to improve the expansion so that something can be said if l = 1: this corresponds to the improvement of the Raabe-Duhamel convergence criterion. This improvement is based on the convergence of a new type of infinite series. These type of series are generalization of the Riemann and Bertrand infinite series.},
     year = {2023}
    }
    

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    AU  - El Hadji Abdoulaye Thiam
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    DO  - 10.11648/j.sjams.20231103.11
    T2  - Science Journal of Applied Mathematics and Statistics
    JF  - Science Journal of Applied Mathematics and Statistics
    JO  - Science Journal of Applied Mathematics and Statistics
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    UR  - https://doi.org/10.11648/j.sjams.20231103.11
    AB  - In Mathematics, convergence tests are methods of testing for the convergence, conditional convergence, absolute convergence, interval of convergence or divergence of an infinite series. There are many criterion for testing the convergence of an infinite series: Cauchy, D’Alembert, Riemann, Bertrand and so one. One of the most important is the Raabe-Duhamel convergence criterion which asserts that: Given an infinite series ∑nun with positive terms un and assuming that the following expansion holds , as n → ∞. Then the series ∑nun converges if λ > 1 and diverges if λ  and  satisfy both the expansion with λ = 1. The first one converges and the second one diverges. The aim of the present paper deals with the convergence of a generalized Riemann-Bertrand infinite series. This will allows us to improve the expansion so that something can be said if l = 1: this corresponds to the improvement of the Raabe-Duhamel convergence criterion. This improvement is based on the convergence of a new type of infinite series. These type of series are generalization of the Riemann and Bertrand infinite series.
    VL  - 11
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Author Information
  • Département de Mathématiques, UFR SET, Université Iba Der THIAM, THIES, Senegal

  • Département de Mathématiques, UFR SET, Université Iba Der THIAM, THIES, Senegal

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