Abstract
The two-parameter method of approximating the sum of a power series in terms of its first three terms of the expansion, which allows one to obtain analytic approximations of various functions, decomposes into a Maclaurin series. As an approximation function of this approximation, it is proposed to use elementary functions constructed in the Nth degree, but with a "compressed" or "stretched" variable x due to the introduction of the numerical factor M (x ≡ ε ∙ m, M ≠ 0) into it. The use of this method makes it possible to significantly increase the range of very accurate approximation of the obtained approximate function with respect to a similar range of the output fragment of a series of three terms. Expressions for both the approximation parameters (M and N) are obtained in a general form and are determined by the coefficients of the second and third terms of the Maclaurin series. Also expressions of both approximation parameters are found for the case if the basis function and the approximant function decompose into the Maclaurin series in even powers of the argument. A number of examples of approximation of functions on the basis of the analysis of power series into which they decompose are given.
Highlights
In the analysis of fundamentally important regularities, from which our understanding of new phenomena develops, the role of analytical methods remains extremely high
If the approximated series is approximated by a new method, for example, within the framework of Newton binomial (in this case, the values of the approximation parameters calculated according to formulas (8): M=5/6, and N=3/5), let’s obtain the approximate formula y(z)= (1+5∙z/6)−0,6, which for z=1 gives an error of less than 0.3 %
Discussion of research results The method for approximating the sum of a power series developed in the article can be used in the case of approximation of functions that decompose into a Taylor series, for example, in the cases when х(0)=d, the power series has the form: y(x)=y(d)+y′(d)∙(x–d)+y′′(d)∙(x–d)2/2+
Summary
In the analysis of fundamentally important regularities, from which our understanding of new phenomena develops, the role of analytical methods remains extremely high. Their role is high in the qualitative determination of the parameters of complex physical phenomena, noted in [1]. It is usually possible to calculate only a few terms of the perturbed decomposition, usually no more than two or three. The resulting rows often converge slowly or even diverge. These several members contain significant information, from which it is necessary to extract all that is possible, is summarized in [2]
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