_{1}

This paper presents the way to make expansion for the next form function: to 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 operators). The paper provides the other kind of solution, except above described theorems.

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

1) The powers of x go down until it reaches

2) The powers of y go up from 0 (

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

5) For each line, the number of product groups is equal to

By using binomial theorem for our case, we obtain next type function [

We can reach the same result by using Fundamental Theorem of Calculus, according it we have [

by means of the addition of integrals

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:

where deltas for

going from it we can get next property of the powers function:

According

Note that upper sign shows the rank of the difference and doesn’t mean power sign. As we can see, according

i | ||||
---|---|---|---|---|

0 | 0 | 1 | 6 | 6 |

1 | 1 | 7 | 12 | 6 |

2 | 8 | 19 | 18 | 6 |

3 | 27 | 37 | 24 | 6 |

4 | 64 | 61 | 30 | 6 |

5 | 125 | 91 | 36 | |

6 | 216 | 127 | ||

7 | 343 |

make conclusion of the next power functions property:

where

Let use sum operator for expression [

Or

Now, we have successful formula, which disperses any natural number

In this section are reviewed the ways to change obtained in previous annex expression [

Expression [

As we can see, iteration limits for [

is not possible. Let change the formula [

Next, give the formula [

Going from expression [

By means of main property of the powers function

According to the above property from the expression [

For

Expression [

By means of Binomial theorem for

According expression [

Let, going from expression [

So, for

oing from it, by means of power function properties, we can only to multiply by x every product of the series, by this way, for

According above method we have right to present function

By means of general

To show changes from binomial theorem, let use other algorithm for

We have right to integrate the

For third derivative we have next equation:

Let derive the

Let be

First derivative is next:

Let calculate the

And for

In case of

So,

and corresponds to binomial expansion. Main difference is adjustable limits of the function [

The paper presented a method of expansion of the function of the form

PetroKolosov, (2016) Series Representation of Power Function. Applied Mathematics,07,327-333. doi: 10.4236/am.2016.73030

Expression [

j = 6 x = Val(Text1.Text) n = Val(Text2.Text) r = 0 For k = 1 To x

Expression [

j = 6 x = Val(Text1.Text) n = Val(Text2.Text) r = 0 For k = 1 To x Step 1 For m = 0 To k − 1 Step 1

Expression [

j = 6

x = Val(Text1.Text)

n = Val(Text2.Text)

r = 0

For k = 1 To x Step 1

For m = 0 To k − 1 Step 1

Next m

Next k

e^{x} Representation:

j = 6

e = 0

x = Val(Text1.Text)

r = Val(Text2.Text)

f = 0

For m = 0 To r Step 1

If m = 0 Then

f = 1

Else

f = f × m

End If

For k = 1 To x

Next k

Next m