Number Theory


Series Representation of Power Function

Authors: Kolosov Petro

In this paper described numerical expansion of natural-valued power function xn, in point x=x_0 where (n, x_0) - natural numbers. Applying numerical methods, that is calculus of finite differences, particular pattern, that is sequence A287326 in OEIS, which shows us necessary items to expand monomial x^3, x∈N is reached and generalized, obtained results are applied to show expansion of power function f(x)=x^n, (x,n)∈N. Received results were compared with solutions according to Newton's Binomial theorem and MacMillan Double Binomial sum. Additionally, in Section 4 exponential function's Exp(x) representation is shown and relation between Pascal's triangle and hypercubes is shown in Section 3. In subsection (2.1) obtained results are applied to show finite difference of power.

Comments: 19 pages, 9 figures, 2 tables, typos and references are revised, results generalized

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Submission history

[v1] 2017-06-09 07:24:09
[v2] 2017-08-01 15:52:40
[v3] 2017-09-07 17:27:09
[v4] 2017-11-23 17:13:00
[v5] 2018-01-12 03:38:22
[v6] 2018-02-17 19:34:49

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