Consequences of Generalized Ratio Tests That Lead to More Efficient Convergence and Divergence Tests

Joseph Granville Gaskin

doi:doi:10.11648/j.pamj.20251401.12

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Consequences of Generalized Ratio Tests That Lead to More Efficient Convergence and Divergence Tests

Joseph Granville Gaskin^*

Published in Pure and Applied Mathematics Journal (Volume 14, Issue 1)

Received: 13 December 2024 Accepted: 6 January 2025 Published: 17 February 2025

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Abstract

The Ratio Test, developed by Jean Le Rond d'Alembert, is a fundamental method for determining the convergence or divergence of an infinite series and is commonly used as a primary test for many series. However, due to its restricted range of applicability, several generalized forms of the Ratio Test have been introduced to extend its usefulness. In this paper, we combine two of the most effective and reliable generalized Ratio Tests to create more efficient convergence tests. To this end, we show that if a positive valued function, f, is defined for all numbers greater than or equal to one, and if the improper integral of the reciprocal of f, over the interval from one to infinity, diverges, then f has a close relationship with a sister function φ. We then show that these paired functions satisfy a remarkable relationship that completely characterizes all monotonically decreasing sequence of terms whose sum diverge. We demonstrate through several examples the ease with which φ can be found if f is known and vice-versa. Next, we combine the generalized Ratio Tests of Dini and Ermakoff by focusing on a ‘thin’ subsequence of the terms of a large category of infinite series to develop other convergence and divergence tests. Furthermore, we refine these tests to produce practical and easier to apply convergence and divergence tests. Lastly, we demonstrate that for many infinite series, one can factor their terms into the product of the reciprocal of f and L. We then show that the limit superior and limit inferior of an expression based on L determines the convergence or divergence of the original series.

Published in	Pure and Applied Mathematics Journal (Volume 14, Issue 1)
DOI	10.11648/j.pamj.20251401.12
Page(s)	8-12
Creative Commons	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), 2025. Published by Science Publishing Group

Keywords

Series, Convergence, Divergence

References

[1]	Bromwich, T. J., An Introduction to the Theory of Infinite Series, Macmillan & Co. Ltd., New York, Second Edition, 1991.
[2]	Bromwich, T. J., An Introduction to the Theory of Infinite Series, Alpha Editions, www.alphaedis.com , (2020).
[3]	Jiri Lebl, Basic Analysis I, Introduction to Real Analysis, Vol 1, 2021, pp 80-86. http://ccw.mit.edu>mit18_100af20_basic_analysis
[4]	Gaskin, J. G., Characterization of a large family of Convergent series that leads to a Rapid Acceleration of Slowly Convergent Logarithmic series, International Journal of Theoretical and Applied Mathematics, 2024, Vol. 10, Issue 3, http:doi.org/10.11648/j.ijtam.20241003.11
[5]	Anton, H, Bivens, I., Davis, S., John Wiley& Sons Inc, pp 309-321.
[6]	Briggs, Cochran, Gillet, Schulz, 3^rd Edition, Calculus Early Transcendentals, Pearson, 2023.
[7]	Stewart, J., Clegg, D. K., Watson, S., Calculus Early Transcendentals, 9^th Edition, 2020, pp 329.
[8]	Thomson B. S., Bruckner, J. B., Bruckner, A. M., Elementary Real Analysis, 2008, pp 416.
[9]	Walter Rudin, Principles of Mathematical Analysis, International Series in Pure and Applied Mathematics, McGraw-Hill, Third Edition, 1976.
[10]	Bartle R. G., The Elements of Real Analysis, Wiley Publishers, 2^nd Edition, pp 259.
[11]	Hunter K. John, An Introduction to Real Analysis, 2014, pp 267. http://www.math.ucdavis.edu>intro>_analysis_pdf
[12]	Thomas, Calculus, 14^th Edition, George B Thomas, Joel R Hass, Christopher Heil, Maurice D. Weir, Pearson, pp 197.
[13]	Trench, William F., Introduction to Real Analysis, p.cm., Library of Congress Cataloging-in-publication Data, 2013, pp 200-233.
[14]	Jiri Lebl, Basic Analysis I, Introduction to Real Analysis, Vol 1, 2021, pp 80-86. http://ccw.mit.edu>mit18_100af20_basic_analysis
[15]	Sayel A. Ali, The m^th Ratio Test, Convergence Tests for Series, American Mathematical Monthly, 2008, Vol 115, pp. 514-524.

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Gaskin, J. G. (2025). Consequences of Generalized Ratio Tests That Lead to More Efficient Convergence and Divergence Tests. Pure and Applied Mathematics Journal, 14(1), 8-12. https://doi.org/10.11648/j.pamj.20251401.12

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ACS Style

Gaskin, J. G. Consequences of Generalized Ratio Tests That Lead to More Efficient Convergence and Divergence Tests. Pure Appl. Math. J. 2025, 14(1), 8-12. doi: 10.11648/j.pamj.20251401.12

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AMA Style

Gaskin JG. Consequences of Generalized Ratio Tests That Lead to More Efficient Convergence and Divergence Tests. Pure Appl Math J. 2025;14(1):8-12. doi: 10.11648/j.pamj.20251401.12

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@article{10.11648/j.pamj.20251401.12,
  author = {Joseph Granville Gaskin},
  title = {Consequences of Generalized Ratio Tests That Lead to More Efficient Convergence and Divergence Tests},
  journal = {Pure and Applied Mathematics Journal},
  volume = {14},
  number = {1},
  pages = {8-12},
  doi = {10.11648/j.pamj.20251401.12},
  url = {https://doi.org/10.11648/j.pamj.20251401.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20251401.12},
  abstract = {The Ratio Test, developed by Jean Le Rond d'Alembert, is a fundamental method for determining the convergence or divergence of an infinite series and is commonly used as a primary test for many series. However, due to its restricted range of applicability, several generalized forms of the Ratio Test have been introduced to extend its usefulness. In this paper, we combine two of the most effective and reliable generalized Ratio Tests to create more efficient convergence tests. To this end, we show that if a positive valued function, f, is defined for all numbers greater than or equal to one, and if the improper integral of the reciprocal of f, over the interval from one to infinity, diverges, then f has a close relationship with a sister function φ. We then show that these paired functions satisfy a remarkable relationship that completely characterizes all monotonically decreasing sequence of terms whose sum diverge. We demonstrate through several examples the ease with which φ can be found if f is known and vice-versa. Next, we combine the generalized Ratio Tests of Dini and Ermakoff by focusing on a ‘thin’ subsequence of the terms of a large category of infinite series to develop other convergence and divergence tests. Furthermore, we refine these tests to produce practical and easier to apply convergence and divergence tests. Lastly, we demonstrate that for many infinite series, one can factor their terms into the product of the reciprocal of f and L. We then show that the limit superior and limit inferior of an expression based on L determines the convergence or divergence of the original series.},
 year = {2025}
}

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TY  - JOUR
T1  - Consequences of Generalized Ratio Tests That Lead to More Efficient Convergence and Divergence Tests
AU  - Joseph Granville Gaskin
Y1  - 2025/02/17
PY  - 2025
N1  - https://doi.org/10.11648/j.pamj.20251401.12
DO  - 10.11648/j.pamj.20251401.12
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 8
EP  - 12
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20251401.12
AB  - The Ratio Test, developed by Jean Le Rond d'Alembert, is a fundamental method for determining the convergence or divergence of an infinite series and is commonly used as a primary test for many series. However, due to its restricted range of applicability, several generalized forms of the Ratio Test have been introduced to extend its usefulness. In this paper, we combine two of the most effective and reliable generalized Ratio Tests to create more efficient convergence tests. To this end, we show that if a positive valued function, f, is defined for all numbers greater than or equal to one, and if the improper integral of the reciprocal of f, over the interval from one to infinity, diverges, then f has a close relationship with a sister function φ. We then show that these paired functions satisfy a remarkable relationship that completely characterizes all monotonically decreasing sequence of terms whose sum diverge. We demonstrate through several examples the ease with which φ can be found if f is known and vice-versa. Next, we combine the generalized Ratio Tests of Dini and Ermakoff by focusing on a ‘thin’ subsequence of the terms of a large category of infinite series to develop other convergence and divergence tests. Furthermore, we refine these tests to produce practical and easier to apply convergence and divergence tests. Lastly, we demonstrate that for many infinite series, one can factor their terms into the product of the reciprocal of f and L. We then show that the limit superior and limit inferior of an expression based on L determines the convergence or divergence of the original series.
VL  - 14
IS  - 1
ER  -

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Author Information

Joseph Granville Gaskin

Department of Mathematics & Physics, Southern University, Baton Rouge, USA

Contact Email

http://orcid.org/0009-0004-2781-647X

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Plain Text BibTeX RIS

APA Style

Gaskin, J. G. (2025). Consequences of Generalized Ratio Tests That Lead to More Efficient Convergence and Divergence Tests. Pure and Applied Mathematics Journal, 14(1), 8-12. https://doi.org/10.11648/j.pamj.20251401.12

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ACS Style

Gaskin, J. G. Consequences of Generalized Ratio Tests That Lead to More Efficient Convergence and Divergence Tests. Pure Appl. Math. J. 2025, 14(1), 8-12. doi: 10.11648/j.pamj.20251401.12

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AMA Style

Gaskin JG. Consequences of Generalized Ratio Tests That Lead to More Efficient Convergence and Divergence Tests. Pure Appl Math J. 2025;14(1):8-12. doi: 10.11648/j.pamj.20251401.12

Copy | Download

@article{10.11648/j.pamj.20251401.12,
  author = {Joseph Granville Gaskin},
  title = {Consequences of Generalized Ratio Tests That Lead to More Efficient Convergence and Divergence Tests},
  journal = {Pure and Applied Mathematics Journal},
  volume = {14},
  number = {1},
  pages = {8-12},
  doi = {10.11648/j.pamj.20251401.12},
  url = {https://doi.org/10.11648/j.pamj.20251401.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20251401.12},
  abstract = {The Ratio Test, developed by Jean Le Rond d'Alembert, is a fundamental method for determining the convergence or divergence of an infinite series and is commonly used as a primary test for many series. However, due to its restricted range of applicability, several generalized forms of the Ratio Test have been introduced to extend its usefulness. In this paper, we combine two of the most effective and reliable generalized Ratio Tests to create more efficient convergence tests. To this end, we show that if a positive valued function, f, is defined for all numbers greater than or equal to one, and if the improper integral of the reciprocal of f, over the interval from one to infinity, diverges, then f has a close relationship with a sister function φ. We then show that these paired functions satisfy a remarkable relationship that completely characterizes all monotonically decreasing sequence of terms whose sum diverge. We demonstrate through several examples the ease with which φ can be found if f is known and vice-versa. Next, we combine the generalized Ratio Tests of Dini and Ermakoff by focusing on a ‘thin’ subsequence of the terms of a large category of infinite series to develop other convergence and divergence tests. Furthermore, we refine these tests to produce practical and easier to apply convergence and divergence tests. Lastly, we demonstrate that for many infinite series, one can factor their terms into the product of the reciprocal of f and L. We then show that the limit superior and limit inferior of an expression based on L determines the convergence or divergence of the original series.},
 year = {2025}
}

Copy | Download

TY  - JOUR
T1  - Consequences of Generalized Ratio Tests That Lead to More Efficient Convergence and Divergence Tests
AU  - Joseph Granville Gaskin
Y1  - 2025/02/17
PY  - 2025
N1  - https://doi.org/10.11648/j.pamj.20251401.12
DO  - 10.11648/j.pamj.20251401.12
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 8
EP  - 12
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20251401.12
AB  - The Ratio Test, developed by Jean Le Rond d'Alembert, is a fundamental method for determining the convergence or divergence of an infinite series and is commonly used as a primary test for many series. However, due to its restricted range of applicability, several generalized forms of the Ratio Test have been introduced to extend its usefulness. In this paper, we combine two of the most effective and reliable generalized Ratio Tests to create more efficient convergence tests. To this end, we show that if a positive valued function, f, is defined for all numbers greater than or equal to one, and if the improper integral of the reciprocal of f, over the interval from one to infinity, diverges, then f has a close relationship with a sister function φ. We then show that these paired functions satisfy a remarkable relationship that completely characterizes all monotonically decreasing sequence of terms whose sum diverge. We demonstrate through several examples the ease with which φ can be found if f is known and vice-versa. Next, we combine the generalized Ratio Tests of Dini and Ermakoff by focusing on a ‘thin’ subsequence of the terms of a large category of infinite series to develop other convergence and divergence tests. Furthermore, we refine these tests to produce practical and easier to apply convergence and divergence tests. Lastly, we demonstrate that for many infinite series, one can factor their terms into the product of the reciprocal of f and L. We then show that the limit superior and limit inferior of an expression based on L determines the convergence or divergence of the original series.
VL  - 14
IS  - 1
ER  -

Copy | Download