A method for recursively generating sequential rational approximations to $\sqrt[n]{k}$

TL;DR

This paper introduces a recursive method using nonlinear difference equations to generate increasingly accurate rational approximations of the n-th root of k, analyzed through dynamical systems theory.

Contribution

It presents a novel recursive approach for approximating roots, analyzed via discrete dynamical systems, avoiding traditional infinitesimal calculus.

Findings

The recursion converges to the correct root under certain conditions.

Convergence is characterized by eigenvector analysis of the associated dynamical system.

The method provides a new perspective on root approximation using difference equations.

Abstract

The goal of this paper is to derive a simple recursion that generates a sequence of fractions approximating $\sqrt[n]{k}$ with increasing accuracy. The recursion is defined in terms of a series of first-order non-linear difference equations and then analyzed as a discrete dynamical system. Convergence behavior is then discussed in the language of initial trajectories and eigenvectors, effectively proving convergence without notions from standard analysis of infinitesimals.