Equivalent series theorem and obtaining some new summable numerical series using fast expansion polynomials

A theorem on equivalent uniformly converging series is proved, which allows one to find a set of new spectra of summable series. The classification of number series into three classes is given. By expanding special polynomials in Fourier series in terms of various fundamental system of functions, it is possible to summarize a wide class of new numerical series that have a convenient classical form of the first class. These polynomials are borrowed from the fast expansions method of the authors of this article.


Introduction
Some summable numerical series obtained in the early and middle of the 20th century are known. Various authors found particular cases; therefore, the series they obtained were published as accompanying results in separate scientific articles and monographs. These particular cases of numerical series are collected in reference books [1,2]. Since then no new summable series have been found. These reference books contain a large number of functional series built with help of complex special functions. Recently, some studies of various properties of numerical series [3][4][5][6] have been conducted. In [7], it was obtained that the order of decreasing of the norm L in the remainder of the Fourier sine series with monotonic coefficients is expressed through the coefficients of the series in the same way as for the series with convex coefficients.
In this work, to obtain new numerical series, fast polynomials are used, which have a simple form; therefore, they allow you to organize several sets of summed spectra of numerical series. It is convenient to use such series to evaluate some functional series. They can also be used as majorant series, in computational practice and theoretical research.

Problem setup
All known summable numerical series are obtained from functional ones, where the variable x is given some characteristic values. Below, the uniformly converging functional series will be written in In this article, when obtaining new numerical series, we will permute the terms of series (1) in some definite order. The question arises about the legality of such a permutation. In this connection, we prove the theorem.
A theorem on equivalent uniformly converging series. Let the series is some arbitrary given natural number, called the numeric period. Then the equality takes place: n n n n n n The penultimate row on the left of (4) is equal to . Equality (4) means that the sum A of the series in the left-hand side is equivalent to one row in the right-hand side. Let us prove the theorem. By the conditions of the theorem, the number of series on the left-hand side of (4) is equal A and each series in (4) converges uniformly. To prove the theorem, we write down the partial sums of these series in (4) as follows: , .
On the left-hand side of (5), each term of the first partial sum is not repeated in the remaining partial sums, and each member of the second partial sum is not repeated in the remaining partial sums. The partial sum of each row on the left side of (5) consists of 0 1 K + terms. The total number of terms in the A rows on the left side is equal to ( ) 0 1 A K + that is equal to the number of the first terms of the partial sum of the row on the right side. Each term of the partial sum on the right-hand side of (5) is contained once in the left-hand side, i.e. equality (5) is satisfied exactly. Since all the series under consideration converge uniformly, the limit of the sum of partial sums of a finite number A of uniformly converging series on the left is equal to the sum of their limits and is equal to the sum of the series on the right: Below, all new summable numerical series will be obtained using the proved theorem. The validity of this theorem can also be verified by direct summing on a computer of the terms of the series under consideration and calculating the corresponding values of the fast polynomials through which the sums are expressed.

Materials and methods
Now we consider fast polynomials and the classification of numerical series. Converging series of numbers can be divided into three classes.
The first class includes numerical series, the sum of which is expressed through elementary functions in a finite form. Such a sum is easily calculated even on a calculator and is convenient for practical use. There are few such numerical series; they are given in reference books [1,2].
The second class includes numerical series, the sum of which is expressed in the final form through special functions, which in turn are expressed by some series. In this case, the calculation of the sum of the series under consideration is replaced by a computer calculation of the series for the special function. The reference books [1,2] mainly show the ranks of the second class.
The third class should include all other converging numerical series, for which the expression of the sum of the series in the final form through elementary or special functions is unknown. Their sum can be calculated on a computer by direct summation of the members of the series.
Below, a set of spectra of number series of the first class will be obtained, which is a significant addition to the general list of similar series.
For these purposes, polynomials borrowed from the method of fast expansions [7] are used, hereinafter called fast polynomials. First, we indicate the logical path leading to the summation of the numerical series below, so as not to create the impression of the random nature of the results obtained.
When using the method of fast expansions, we represent an arbitrary smooth function ( ) Then ( ) In (14) ( ) ( ) 2 2 , m m P x Q x are the fast polynomials calculated by the following recurrent integral formulas The polynomials ( ) ( ) , m m P x Q x can also be calculated from the solution of boundary value differential problems with zero Dirichlet boundary conditions, for which they were created [7, 8]: with odd indices are used: 7 5 3 7 31 127 9! 6 7! 60 6! 18 7! 604800 and ( ) Fast polynomials expressions (19) and (20) do not depend on the order of the boundary function. They have special properties; one of them is used to obtain new summable wonderful numerical series by expanding into corresponding Fourier series.
Let us consider construction of numerical series using fast polynomials. Numerical series will be constructed by expanding polynomials (19) in Here the right side can be represented by the following rows: This equality is a consequence of formula (11) at 6 A = , which after simplification takes the form: We have the spectrum of numerical series by setting 1 3 x = in (24): We obtain a numerical series at 2 A = from (24) at 1 If we take 1 3 x = in (30), we get a series for 3 A = :  1  1  2  1  1  1  ,  3  3  3 2  3 1   1  1  1  2  1  1  .  3  3  3 We supplement the set of numerical series (25) Here ( ) ( )  1  1  1  1,  ,  ,  ,  2  6  2  1  1  1  1  1  5  ,  ,  24  6  3  120  12  24  1  1  1  2 ,  720  120  18  15  1  1  5  61 .  5040  240  144 720 In this case, to obtain new numerical series, we also expand ( )   In the series (38), to obtain new numerical series, the variable x is assumed to be equal 1 ; 2 3 ; 1 2 ; 1 3 , for which the number series are obtained in a convenient form. If we take 0 x = , then expressions (38) turn into identities. For 1 x = , we find the following two spectra of number rows: we have: Among the obtained series, the expressions for (27) Expression (26) can also be simplified. For this, we exclude the second sum from (26) using the second series from (39): For further simplifications, we exclude the second sum from the first equality (32) using the first equality (31), and from the second equality (32) we exclude the second sum using the second equality (31): To simplify equalities (32), we exclude the first sums in these expressions using two equalities (31): Equalities (42) after simplification by means of two series from (39) are also simplified:

Results and discussion
As an example of a uniformly converging functional series, let us consider the series [1,2] ( )