Approximation of the Sum of a Power Series by Its First Four Terms

Konstantin Ludanov

doi:doi:10.11648/j.ajmp.20231202.12

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Approximation of the Sum of a Power Series by Its First Four Terms

Konstantin Ludanov

Published in American Journal of Modern Physics (Volume 12, Issue 2)

Received: 16 May 2023 Accepted: 5 June 2023 Published: 20 June 2023

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Abstract

The paper develops a three-parameter method for approximating the sum of the McLaurin series by its first four expansion terms, which allows obtaining analytical approximations for functions that are expanded into a power series. The expressions for the approximation parameters (a, b, c) of the exact sum ∑(S) of the geometric power series-base are obtained in general form and are determined by the coefficients at the second (A), third (B), and fourth (C) terms of the McLaurin series. For series that converge rapidly {their coefficients satisfy the inequality (а_n)²≥(a_n‒1×a_n+1)}, the new method gives the real values of the sum ∑(S), and for series that converge slowly {for them (а_n)²<(a_n‒1×a_n+1)}, the method gives the complex-conjugate roots of the parameters of their sum ∑(S). The paper presents examples of approximate determination of series sums by both three-parameter and two-parameter methods based on the analysis of series coefficients. The accuracy of the two- and three-parameter methods of approximation of ∑(S) is evaluated on the basis of determining the approximate sums of known numerical series (for the number , the number e, etc.). The new three-parameter method was used to approximate the sum of a series whose first terms were obtained by Lord Rayleigh when refining the method of determining the capillary complex of a liquid by the capillary rise method.

Published in	American Journal of Modern Physics (Volume 12, Issue 2)
DOI	10.11648/j.ajmp.20231202.12
Page(s)	21-29
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

Sum of the Series Approximation, Three-Parameter Approximation, McLaurin's Sum Series Approximation, Sum of Numerical Series Estimation, Rayleigh's Series Decomposition, Rayleigh's Sum Series Calculation

References

[1]	Cole J. Perturbation Methods in Applied Mathematics. - M: MIR, 1972.
[2]	Van Dyck M. Perturbation methods in fluid mechanics. - Nauka, 1967.
[3]	Andrianov I. V., Manevich L. I. Asymptotic methods and physical theories. - M: "ZNANIE". Physics series, 2/1989.
[4]	Prudnikov A. P., Brychkov Y. A., Marichev O. I. INTEGRALS & SERIES. Elementary functions. - M.: Nauka. GR FML, 1981.
[5]	Solomentsev, E. D. (2001), "POWER SERIES", Encyclopedia of Mathematics, EMS Press.
[6]	Baker J., Graves Maurice P. Padé approximations. - M: MIR, 1986.
[7]	Vinogradov, V. N., Gay, E. V., Rabotnov, N. S. (1987). Analytical approximation of data in nuclear and neutron physics. M: Energoatomizdat, 128.
[8]	Ludanov K. I. Method for obtaining approximated formulas // Scientific journal "EURICA: Physics and Engineering", 2018, 2 (15), pp. 72-78.
[9]	Dwight G. B. Tables of integrals and other mathematical formulas. - M: Nauka, 1964.
[10]	Tsypkin A. G., Tsypkin G. G. Mathematical formulas. - M.: Nauka. FML, 1985.
[11]	Korn G., Korn T. MATHEMATICS GUIDE. Definitions, theorems, formulas. - M.: Nauka. GR FML, 1975.
[12]	Adamson A. Physical chemistry of surfaces. - M.: MIR. 1979.
[13]	Lord Rayleigh (Strutt J. W.), Proc. Roy. Soc. (London), A92, 184 (1915).
[14]	Ludanov Konstantin. (2019) Method of joint determination of the capillary complex and kinematic viscosity. International Journal IJRTEM. Vol. 3, Issue 7, No. 11/12, PP 34-38.
[15]	Konstantin Ludanov. Methods for Joint Determination of the Surface Tension Coefficient of a Liquid and the Contact Angle of Wetting the Hard Surface. World Journal of Applied Physics. Vol. 5, No. 3, 2020, pp. 39-42. doi: 10.11648/j.wjap.20200503.12.

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Konstantin Ludanov. (2023). Approximation of the Sum of a Power Series by Its First Four Terms. American Journal of Modern Physics, 12(2), 21-29. https://doi.org/10.11648/j.ajmp.20231202.12

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ACS Style

Konstantin Ludanov. Approximation of the Sum of a Power Series by Its First Four Terms. Am. J. Mod. Phys. 2023, 12(2), 21-29. doi: 10.11648/j.ajmp.20231202.12

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AMA Style

Konstantin Ludanov. Approximation of the Sum of a Power Series by Its First Four Terms. Am J Mod Phys. 2023;12(2):21-29. doi: 10.11648/j.ajmp.20231202.12

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@article{10.11648/j.ajmp.20231202.12,
  author = {Konstantin Ludanov},
  title = {Approximation of the Sum of a Power Series by Its First Four Terms},
  journal = {American Journal of Modern Physics},
  volume = {12},
  number = {2},
  pages = {21-29},
  doi = {10.11648/j.ajmp.20231202.12},
  url = {https://doi.org/10.11648/j.ajmp.20231202.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajmp.20231202.12},
  abstract = {The paper develops a three-parameter method for approximating the sum of the McLaurin series by its first four expansion terms, which allows obtaining analytical approximations for functions that are expanded into a power series. The expressions for the approximation parameters (a, b, c) of the exact sum ∑(S) of the geometric power series-base are obtained in general form and are determined by the coefficients at the second (A), third (B), and fourth (C) terms of the McLaurin series. For series that converge rapidly {their coefficients satisfy the inequality (аn)2≥(an‒1×an+1)}, the new method gives the real values of the sum ∑(S), and for series that converge slowly {for them (аn)2an‒1×an+1)}, the method gives the complex-conjugate roots of the parameters of their sum ∑(S). The paper presents examples of approximate determination of series sums by both three-parameter and two-parameter methods based on the analysis of series coefficients. The accuracy of the two- and three-parameter methods of approximation of ∑(S) is evaluated on the basis of determining the approximate sums of known numerical series (for the number , the number e, etc.). The new three-parameter method was used to approximate the sum of a series whose first terms were obtained by Lord Rayleigh when refining the method of determining the capillary complex of a liquid by the capillary rise method.},
 year = {2023}
}

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TY  - JOUR
T1  - Approximation of the Sum of a Power Series by Its First Four Terms
AU  - Konstantin Ludanov
Y1  - 2023/06/20
PY  - 2023
N1  - https://doi.org/10.11648/j.ajmp.20231202.12
DO  - 10.11648/j.ajmp.20231202.12
T2  - American Journal of Modern Physics
JF  - American Journal of Modern Physics
JO  - American Journal of Modern Physics
SP  - 21
EP  - 29
PB  - Science Publishing Group
SN  - 2326-8891
UR  - https://doi.org/10.11648/j.ajmp.20231202.12
AB  - The paper develops a three-parameter method for approximating the sum of the McLaurin series by its first four expansion terms, which allows obtaining analytical approximations for functions that are expanded into a power series. The expressions for the approximation parameters (a, b, c) of the exact sum ∑(S) of the geometric power series-base are obtained in general form and are determined by the coefficients at the second (A), third (B), and fourth (C) terms of the McLaurin series. For series that converge rapidly {their coefficients satisfy the inequality (аn)2≥(an‒1×an+1)}, the new method gives the real values of the sum ∑(S), and for series that converge slowly {for them (аn)2an‒1×an+1)}, the method gives the complex-conjugate roots of the parameters of their sum ∑(S). The paper presents examples of approximate determination of series sums by both three-parameter and two-parameter methods based on the analysis of series coefficients. The accuracy of the two- and three-parameter methods of approximation of ∑(S) is evaluated on the basis of determining the approximate sums of known numerical series (for the number , the number e, etc.). The new three-parameter method was used to approximate the sum of a series whose first terms were obtained by Lord Rayleigh when refining the method of determining the capillary complex of a liquid by the capillary rise method.
VL  - 12
IS  - 2
ER  -

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Author Information

Konstantin Ludanov

Chair of Thermophysics, Faculty of Thermal Power Engineering of NTUU “KPI”, Kiev, Ukraine

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APA Style

Konstantin Ludanov. (2023). Approximation of the Sum of a Power Series by Its First Four Terms. American Journal of Modern Physics, 12(2), 21-29. https://doi.org/10.11648/j.ajmp.20231202.12

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ACS Style

Konstantin Ludanov. Approximation of the Sum of a Power Series by Its First Four Terms. Am. J. Mod. Phys. 2023, 12(2), 21-29. doi: 10.11648/j.ajmp.20231202.12

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AMA Style

Konstantin Ludanov. Approximation of the Sum of a Power Series by Its First Four Terms. Am J Mod Phys. 2023;12(2):21-29. doi: 10.11648/j.ajmp.20231202.12

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@article{10.11648/j.ajmp.20231202.12,
  author = {Konstantin Ludanov},
  title = {Approximation of the Sum of a Power Series by Its First Four Terms},
  journal = {American Journal of Modern Physics},
  volume = {12},
  number = {2},
  pages = {21-29},
  doi = {10.11648/j.ajmp.20231202.12},
  url = {https://doi.org/10.11648/j.ajmp.20231202.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajmp.20231202.12},
  abstract = {The paper develops a three-parameter method for approximating the sum of the McLaurin series by its first four expansion terms, which allows obtaining analytical approximations for functions that are expanded into a power series. The expressions for the approximation parameters (a, b, c) of the exact sum ∑(S) of the geometric power series-base are obtained in general form and are determined by the coefficients at the second (A), third (B), and fourth (C) terms of the McLaurin series. For series that converge rapidly {their coefficients satisfy the inequality (аn)2≥(an‒1×an+1)}, the new method gives the real values of the sum ∑(S), and for series that converge slowly {for them (аn)2an‒1×an+1)}, the method gives the complex-conjugate roots of the parameters of their sum ∑(S). The paper presents examples of approximate determination of series sums by both three-parameter and two-parameter methods based on the analysis of series coefficients. The accuracy of the two- and three-parameter methods of approximation of ∑(S) is evaluated on the basis of determining the approximate sums of known numerical series (for the number , the number e, etc.). The new three-parameter method was used to approximate the sum of a series whose first terms were obtained by Lord Rayleigh when refining the method of determining the capillary complex of a liquid by the capillary rise method.},
 year = {2023}
}

Copy | Download

TY  - JOUR
T1  - Approximation of the Sum of a Power Series by Its First Four Terms
AU  - Konstantin Ludanov
Y1  - 2023/06/20
PY  - 2023
N1  - https://doi.org/10.11648/j.ajmp.20231202.12
DO  - 10.11648/j.ajmp.20231202.12
T2  - American Journal of Modern Physics
JF  - American Journal of Modern Physics
JO  - American Journal of Modern Physics
SP  - 21
EP  - 29
PB  - Science Publishing Group
SN  - 2326-8891
UR  - https://doi.org/10.11648/j.ajmp.20231202.12
AB  - The paper develops a three-parameter method for approximating the sum of the McLaurin series by its first four expansion terms, which allows obtaining analytical approximations for functions that are expanded into a power series. The expressions for the approximation parameters (a, b, c) of the exact sum ∑(S) of the geometric power series-base are obtained in general form and are determined by the coefficients at the second (A), third (B), and fourth (C) terms of the McLaurin series. For series that converge rapidly {their coefficients satisfy the inequality (аn)2≥(an‒1×an+1)}, the new method gives the real values of the sum ∑(S), and for series that converge slowly {for them (аn)2an‒1×an+1)}, the method gives the complex-conjugate roots of the parameters of their sum ∑(S). The paper presents examples of approximate determination of series sums by both three-parameter and two-parameter methods based on the analysis of series coefficients. The accuracy of the two- and three-parameter methods of approximation of ∑(S) is evaluated on the basis of determining the approximate sums of known numerical series (for the number , the number e, etc.). The new three-parameter method was used to approximate the sum of a series whose first terms were obtained by Lord Rayleigh when refining the method of determining the capillary complex of a liquid by the capillary rise method.
VL  - 12
IS  - 2
ER  -

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