Characterization of a Large Family of Convergent Series That Leads to a Rapid Acceleration of Slowly Convergent Logarithmic Series

Joseph Gaskin

doi:doi:10.11648/j.ijtam.20241003.11

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Characterization of a Large Family of Convergent Series That Leads to a Rapid Acceleration of Slowly Convergent Logarithmic Series

Joseph Gaskin^*

Published in International Journal of Theoretical and Applied Mathematics (Volume 10, Issue 3)

Received: 3 September 2024 Accepted: 19 September 2024 Published: 10 October 2024

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Abstract

Logarithmic series are known to have a very slow rate of convergence. For example, it takes more than the first 20,000 terms of the sum of the reciprocals of squares of the natural numbers to attain 5 decimal places of accuracy. In this paper, I will devise an acceleration scheme that will yield the same level of accuracy with just the first 400 terms of that power series. To accomplish this, I establish a relationship between all monotonically decreasing sequence of positive terms whose sum converges, a positive number ρ and a differentiable function φ. Then, I use ρ and φ to define the T_{φ, ρ}transformations on the partial sums of any convergent series. Furthermore, I prove that these T_{φ, ρ}transformations yield a rapid rate of convergence for many slowly convergent logarithmic series. Finaly, I provide several examples on how to compute φ if one is given the convergent series of decreasing, positive terms.

Published in	International Journal of Theoretical and Applied Mathematics (Volume 10, Issue 3)
DOI	10.11648/j.ijtam.20241003.11
Page(s)	33-37
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), 2024. Published by Science Publishing Group

Keywords

Series, Accelerators, Logarithmic, Convergence

References

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[2]	Bromwich, T. J., An Introduction to the Theory of Infinite Series, Alpha Editions, www.alphaedis.com (2020).
[3]	Belghaba K., On the Transformation T_+m due to Gray and Clark, Journal of Mathematics and Statistics, Vol. 3, https://doi.org/10.3844/jmssp,2007.243.248
[4]	H. L. Gray, and W. D. Clark, “On a Class of Nonlinear Transformation and their Applications to the Evaluation of Infinite Series,” Journal of Research of the National Bureau of Standards-B. Mathematical Sciences, Vol. 73B, No. 3, July-September 1969.
[5]	J. P. Delahaye and B. Germain-Bonne, SIAM Journal on Numerical Analysis, Vol. 19, No. 4, 1982, pp 840-844.
[6]	C. Brezinski, Convergence acceleration during the 20^th century, Journal of Computational and Applied Mathematics, Volume 122, Issues 1-2, October 2000, pp 1-2.
[7]	d’AsperemontA, Scieur D, Taylor A2021, Acceleration Methods, Foundations and Trends (R) in Optimization, 5(1-2), pp1-245 nowpublishers.com
[8]	Briggs, Cochran, Gillet, Shultz, Calculus Early Transcendentals, 3^rd Edition, ISBN13: 978-0-13-476684-3, 2019, pp 674.
[9]	Howard Anton, I. C. Bivens, Stephen Davis, Calculus, 12^th Edition, 2021, ISBN-13:978-1119778127, ISBN-10:1119778123, pp 290.
[10]	Andrew H. Van Tuyl, Acceleration of Convergence of a family of logarithmically convergent sequences, Mathematics of Computation, Vol 63, Number 207, July 1994, pp 229.
[11]	S. Mukherjee, B-S Hua, N. Umetani, D. Meister, Neural sequence transformation, Computer Graphics Forum, The Eurographics Association and John Wiley & Sons Ltd, 2021, pp 132.
[12]	Khan SA, International Journal of Applied Mathematics, ISBN: 314-8060, Volume 33, 2020. https://doi.org/10.1273/ijam.v33i2.6
[13]	Robert G. Bartle, Donald R. Sherbert, Introduction to Real Analysis, 4^th Edition, ISBN-13: 978-0471433316, 2011, pp 400-418.
[14]	James Stewart, Daniel K. Clegg, Saleem Watson, Calculus: Early Transcendentals, ISBN-13: 978-1337624183, 9^th Edition, April 2020, Chapter 7.8, pp 534.
[15]	Britannica, The Editors Encyclopedia, “Infinite series”, Encyclopedia Britannica, August 2024.

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APA Style

Gaskin, J. (2024). 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, 10(3), 33-37. https://doi.org/10.11648/j.ijtam.20241003.11

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ACS Style

Gaskin, J. Characterization of a Large Family of Convergent Series That Leads to a Rapid Acceleration of Slowly Convergent Logarithmic Series. Int. J. Theor. Appl. Math. 2024, 10(3), 33-37. doi: 10.11648/j.ijtam.20241003.11

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AMA Style

Gaskin J. Characterization of a Large Family of Convergent Series That Leads to a Rapid Acceleration of Slowly Convergent Logarithmic Series. Int J Theor Appl Math. 2024;10(3):33-37. doi: 10.11648/j.ijtam.20241003.11

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@article{10.11648/j.ijtam.20241003.11,
  author = {Joseph Gaskin},
  title = {Characterization of a Large Family of Convergent Series That Leads to a Rapid Acceleration of Slowly Convergent Logarithmic Series
},
  journal = {International Journal of Theoretical and Applied Mathematics},
  volume = {10},
  number = {3},
  pages = {33-37},
  doi = {10.11648/j.ijtam.20241003.11},
  url = {https://doi.org/10.11648/j.ijtam.20241003.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijtam.20241003.11},
  abstract = {Logarithmic series are known to have a very slow rate of convergence. For example, it takes more than the first 20,000 terms of the sum of the reciprocals of squares of the natural numbers to attain 5 decimal places of accuracy. In this paper, I will devise an acceleration scheme that will yield the same level of accuracy with just the first 400 terms of that power series. To accomplish this, I establish a relationship between all monotonically decreasing sequence of positive terms whose sum converges, a positive number ρ and a differentiable function φ. Then, I use ρ and φ to define the Tφ, ρ transformations on the partial sums of any convergent series. Furthermore, I prove that these Tφ, ρ transformations yield a rapid rate of convergence for many slowly convergent logarithmic series. Finaly, I provide several examples on how to compute φ if one is given the convergent series of decreasing, positive terms.
},
 year = {2024}
}

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TY  - JOUR
T1  - Characterization of a Large Family of Convergent Series That Leads to a Rapid Acceleration of Slowly Convergent Logarithmic Series

AU  - Joseph Gaskin
Y1  - 2024/10/10
PY  - 2024
N1  - https://doi.org/10.11648/j.ijtam.20241003.11
DO  - 10.11648/j.ijtam.20241003.11
T2  - International Journal of Theoretical and Applied Mathematics
JF  - International Journal of Theoretical and Applied Mathematics
JO  - International Journal of Theoretical and Applied Mathematics
SP  - 33
EP  - 37
PB  - Science Publishing Group
SN  - 2575-5080
UR  - https://doi.org/10.11648/j.ijtam.20241003.11
AB  - Logarithmic series are known to have a very slow rate of convergence. For example, it takes more than the first 20,000 terms of the sum of the reciprocals of squares of the natural numbers to attain 5 decimal places of accuracy. In this paper, I will devise an acceleration scheme that will yield the same level of accuracy with just the first 400 terms of that power series. To accomplish this, I establish a relationship between all monotonically decreasing sequence of positive terms whose sum converges, a positive number ρ and a differentiable function φ. Then, I use ρ and φ to define the Tφ, ρ transformations on the partial sums of any convergent series. Furthermore, I prove that these Tφ, ρ transformations yield a rapid rate of convergence for many slowly convergent logarithmic series. Finaly, I provide several examples on how to compute φ if one is given the convergent series of decreasing, positive terms.

VL  - 10
IS  - 3
ER  -

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Author Information

Joseph Gaskin

Department of Mathematics and Physics, Southern University and A&M College, Baton Rouge, USA

Contact Email

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

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Plain Text BibTeX RIS

APA Style

Gaskin, J. (2024). 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, 10(3), 33-37. https://doi.org/10.11648/j.ijtam.20241003.11

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ACS Style

Gaskin, J. Characterization of a Large Family of Convergent Series That Leads to a Rapid Acceleration of Slowly Convergent Logarithmic Series. Int. J. Theor. Appl. Math. 2024, 10(3), 33-37. doi: 10.11648/j.ijtam.20241003.11

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AMA Style

Gaskin J. Characterization of a Large Family of Convergent Series That Leads to a Rapid Acceleration of Slowly Convergent Logarithmic Series. Int J Theor Appl Math. 2024;10(3):33-37. doi: 10.11648/j.ijtam.20241003.11

Copy | Download

@article{10.11648/j.ijtam.20241003.11,
  author = {Joseph Gaskin},
  title = {Characterization of a Large Family of Convergent Series That Leads to a Rapid Acceleration of Slowly Convergent Logarithmic Series
},
  journal = {International Journal of Theoretical and Applied Mathematics},
  volume = {10},
  number = {3},
  pages = {33-37},
  doi = {10.11648/j.ijtam.20241003.11},
  url = {https://doi.org/10.11648/j.ijtam.20241003.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijtam.20241003.11},
  abstract = {Logarithmic series are known to have a very slow rate of convergence. For example, it takes more than the first 20,000 terms of the sum of the reciprocals of squares of the natural numbers to attain 5 decimal places of accuracy. In this paper, I will devise an acceleration scheme that will yield the same level of accuracy with just the first 400 terms of that power series. To accomplish this, I establish a relationship between all monotonically decreasing sequence of positive terms whose sum converges, a positive number ρ and a differentiable function φ. Then, I use ρ and φ to define the Tφ, ρ transformations on the partial sums of any convergent series. Furthermore, I prove that these Tφ, ρ transformations yield a rapid rate of convergence for many slowly convergent logarithmic series. Finaly, I provide several examples on how to compute φ if one is given the convergent series of decreasing, positive terms.
},
 year = {2024}
}

Copy | Download

TY  - JOUR
T1  - Characterization of a Large Family of Convergent Series That Leads to a Rapid Acceleration of Slowly Convergent Logarithmic Series

AU  - Joseph Gaskin
Y1  - 2024/10/10
PY  - 2024
N1  - https://doi.org/10.11648/j.ijtam.20241003.11
DO  - 10.11648/j.ijtam.20241003.11
T2  - International Journal of Theoretical and Applied Mathematics
JF  - International Journal of Theoretical and Applied Mathematics
JO  - International Journal of Theoretical and Applied Mathematics
SP  - 33
EP  - 37
PB  - Science Publishing Group
SN  - 2575-5080
UR  - https://doi.org/10.11648/j.ijtam.20241003.11
AB  - Logarithmic series are known to have a very slow rate of convergence. For example, it takes more than the first 20,000 terms of the sum of the reciprocals of squares of the natural numbers to attain 5 decimal places of accuracy. In this paper, I will devise an acceleration scheme that will yield the same level of accuracy with just the first 400 terms of that power series. To accomplish this, I establish a relationship between all monotonically decreasing sequence of positive terms whose sum converges, a positive number ρ and a differentiable function φ. Then, I use ρ and φ to define the Tφ, ρ transformations on the partial sums of any convergent series. Furthermore, I prove that these Tφ, ρ transformations yield a rapid rate of convergence for many slowly convergent logarithmic series. Finaly, I provide several examples on how to compute φ if one is given the convergent series of decreasing, positive terms.

VL  - 10
IS  - 3
ER  -

Copy | Download