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

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)2<(an‒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.

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

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.

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

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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