Series Representation of Power Function

Kolosov Petro (1)

1 Odessa National I.I.Mechnikov University

This paper presents the way to make expansion for the next form function: $y=x^n, \ \forall(x,n) \in {\mathbb{N}}$ to the numerical series. The most widely used methods to solve this problem are Newton’s Binomial Theorem and Fundamental Theorem of Calculus (that is, derivative and integral are inverse operators). The paper provides the other kind of solution, based on induction from particular to general case, except above described theorems. \ \ \ Keywords: power, power function, monomial, polynomial, power series, third power, series, finite difference, divided difference, high order finite difference, derivative, binomial coefficient, binomial theorem, Newton's binomial theorem, binomial expansion, n-th difference of n-th power, number theory, cubic number, cube, Euler number, exponential function, Pascal triangle, Pascal’s triangle, mathematics, math, maths, science, arxiv, preprint, наука, математика

Mots clés: binomial expansion; n-th difference of n-th power; Euler number; Pascal triangle; polynomial; power series; high order finite difference; finite difference; monomial; power function; power; Newton's binomial theorem; Pascal's triangle; number theory; cubic number; cube; Exponential function; binomial coefficient; series; Mathematics; Science; Preprint; arxiv; binomial theorem; divided difference; derivative; Math; Maths

Mots clés: Математика; Наука

Domaines: Mathématiques [math]; Théorie des nombres [math.NT]; Analyse classique [math.CA]

Kolosov Petro. Series Representation of Power Function. 2017. ⟨hal-01283042v5⟩