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Research Article |

Improvement of the Raabe-Duhamel Convergence Criterion Generalized

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.

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

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.

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

Copyright © 2023 Authors retain the copyright of this article.
This article is an open access article distributed under the Creative Commons Attribution License ( which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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