Preprint Article Version 4 Preserved in Portico This version is not peer-reviewed

# Series Representation of Power Function

Version 1 : Received: 24 November 2017 / Approved: 24 November 2017 / Online: 24 November 2017 (05:15:26 CET)
Version 2 : Received: 12 January 2018 / Approved: 18 January 2018 / Online: 18 January 2018 (03:37:52 CET)
Version 3 : Received: 18 February 2018 / Approved: 19 February 2018 / Online: 19 February 2018 (16:42:41 CET)
Version 4 : Received: 8 May 2018 / Approved: 9 May 2018 / Online: 9 May 2018 (06:31:05 CEST)
Version 5 : Received: 28 May 2018 / Approved: 28 May 2018 / Online: 28 May 2018 (08:26:14 CEST)
Version 6 : Received: 16 August 2018 / Approved: 17 August 2018 / Online: 17 August 2018 (11:10:47 CEST)

How to cite: Kolosov, P. Series Representation of Power Function. Preprints 2017, 2017110157. https://doi.org/10.20944/preprints201711.0157.v4 Kolosov, P. Series Representation of Power Function. Preprints 2017, 2017110157. https://doi.org/10.20944/preprints201711.0157.v4

## Abstract

In this paper described numerical expansion of natural-valued power function xn, in point x = x0, where n; x0 - positive integers. Applying numerical methods, the calculus of nite di erences, particular pattern, that is sequence A287326 in OEIS, which shows the expansion of perfect cube n as row sum over k; 0 ≤ k ≤ n − 1 is generalized, obtained results are applied to show expansion of monomial n2m+1; m = 0; 1; 2; ..., N. Additionally, relation between Faulhaber's sum nm and nite di erences of power are shown in section 4.

## Keywords

power function; binomial coefficient; binomial theorem; finite difference; perfect cube; Pascal's triangle; series representation; binomial sum; binomial identity; binomial distribution; hypercube

## Subject

Computer Science and Mathematics, Algebra and Number Theory