Series Representation of Power Function

Abstract : In this paper described numerical expansion of natural-valued power function $x^n$, in point $x=x_0$ where $n, \ x_0$ - natural numbers. Applying numerical methods, that is calculus of finite differences, namely, discrete case of Binomial expansion is reached. Received results were compared with solutions according to Newton’s Binomial theorem and MacMillan Double Binomial sum. Additionally, in section 4 exponential function’s $e^x$ representation is shown.
Keywords : petro.kolosov.9 Kolosov_Petro kolosov.petro petrokolosov kolosov-petro petro-kolosov KolosovP kolosov_p_1 Kolosov Kolosov Petro Petro Kolosov Maclaurin Series Taylor polynomial Taylor series Taylor formula Taylor theorem Polynomial expansion Series representation Analytic function Taylor's polynomial Taylor's series Taylor's formula Taylor's theorem Series Expansion 0000-0002-6544-8880 Open access arXiv.org Open science Preprint arXiv h-difference Quantum variatoinal calculus Quantum algebra Qunatum calculus Hypergeometric series Hypergeometric function Time Scale Calculus Power quantum calculus Quantum difference q-difference Quantum calculus h-calculus q-calculus Jackson derivative Monomial Power function Polynomial Power series (mathematics) Power (mathematics) Exponential function Exponentiation Cube (Algebra) Mathematical Series Euler number Finite difference Diophantine equations Perfect cube Divided difference Derivative Ordinary differential equation High order finite difference Partial differential equation Calculus of variations Partial difference Partial derivative High order derivative Differential calculus Binomial coefficient Central Finite difference Backward Finite Difference Finite difference coefficient Forward Finite Difference Numerical Differentiation Difference Equations Finite differences Differentiation Derivatives Binomial Series Pascal’s triangle Newton's interpolation formula Pascal triangle Binomial theorem Multinomial theorem Binomial expansion Calculus Differential equations Binomial Sum Newton's Binomial Theorem Maths Mathematics Math Algebra Numerical analysis Science Number theory Mathematical analysis Functional analysis STEM Numercal methods General Mathematics Analysis of PDEs Classical Analysis and ODEs Applied Mathematics Discrete Mathematics q-derivative
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11 pages, 5 figures, major revisions of sections, abstract shortened and detailed, MSC classifica.. 2017
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Kolosov Petro. Series Representation of Power Function. 11 pages, 5 figures, major revisions of sections, abstract shortened and detailed, MSC classifica.. 2017. 〈hal-01283042v6〉

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