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.
DOI | 10.11648/j.ajmp.20231202.12 |
Published in | American Journal of Modern Physics (Volume 12, Issue 2, March 2023) |
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 |
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
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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
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
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
@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} }
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 -