Series Representation of Power Function

This paper presents the way to make expansion for the next form function: the numerical series. The most widely used methods to solve this problem are Newtons Binomial Theorem and Fundamental Theorem of Calculus (that is, derivative and integral are inverse oper-ators). The paper provides the other kind of solution, except above described theorems.


Introduction
Let basically describe Newtons Binomial Theorem and Fundamental Theorem of Calculus and some their properties. In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial. According to the theorem, it is possible to expand the power ( ) 3) The n-th row of the Pascal's Triangle will be the coefficients of the expanded binomial. 4) For each line, the number of products (i.e. the sum of the coefficients) is equal to 1. x + 5) For each line, the number of product groups is equal to 2 n . By using binomial theorem for our case, we obtain next type function [1]: For presented in this paper method, the properties of binomial theorem are not corresponded and prime function (i.e function, which we use with sum operator) has the recursion structure for x basic view is the next: . Below is represented theoretical algorithm deducing such a function, which, when substituted to the sum operator, with some k number of iterations, returns the correct value of a number x ∈  to power 3 n = . The main idea that the law for basic elements distribution of the value to third powers seen in finding the n-rank difference (n-rank difference is written as n ∆ ) between nearest two items 3 3 1 , Note that upper sign shows the rank of the difference and doesn't mean power sign. As we can see, according Table 1, the values of third rank difference are equal to 3! and constant for each i. Going from it, we can to Table 1. Numbers according third power.
and constant for each i. For Table 1, the delta functions are corresponding to next expressions: [3], we obtain [4]: Now, we have successful formula, which disperses any natural number 3 x to the numerical series (this example shows only expansion for any number x ∈  to third powers, but this method works for floats numbers also, it depends of start set of numbers, functions form also depends of the chosen set, step i x ∆ between numbers should be constant every time). Going from it, the next annex shows change over function to the range ,3 x n ∈ < ∈  .

Change over to Higher Powers Expression
In this section are reviewed the ways to change obtained in previous annex expression [4] to higher powers i.e   ( ) n y x x = ≥ expression takes the next form [9]: ( ) has the next property (as well right for [4] [6]- [8]):

e x Representation
According above method we have right to present function e x y = the follow view (as the exponential function is the infinite sum of powers of x divided by value of factorial according to iteration step):

Conclusion
The paper presented a method of expansion of the function of the form , , n y x x n = ∈ ∈   to the numerical series. The disadvantages of this method are sophisticated form of expression and the complexity of calculating the value of these expressions of the some variables. Advantage of this method is the possibility of the successful application of this method in the solution of some problems in number theory, the theory of series, due to the differences from the common theory, displayed the difference from binomial expansion, presented example for exponential function representation by means of method from Section 2. The paper doesn't consist the all combinations of power function representation (by means of the function [7] property and transformation [5]). In the Application 1 are shown program codes for the most important expressions (by authors' opinion). Future research in this direction could result the success polynomial kind expansion.