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 ∑_{n}u_{n} with positive terms u_{n }and assuming that the following expansion holds , as n → ∞. Then the series ∑_{n}u_{n} 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.
Published in | Science Journal of Applied Mathematics and Statistics (Volume 11, Issue 3) |
DOI | 10.11648/j.sjams.20231103.11 |
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. |
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Copyright © The Author(s), 2023. Published by Science Publishing Group |
Infinite Series, Raabe-Duhamel Convergence Criterion, Riemann and Bertrand Infinite Series
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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
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
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
@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} }
TY - JOUR T1 - Improvement of the Raabe-Duhamel Convergence Criterion Generalized AU - El Hadji Abdoulaye Thiam AU - Papa Moussa Niang Y1 - 2023/11/16 PY - 2023 N1 - https://doi.org/10.11648/j.sjams.20231103.11 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 SP - 44 EP - 47 PB - Science Publishing Group SN - 2376-9513 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 IS - 3 ER -