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

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.

References in corpus (3)

• A Lanczos Method for Approximating Composite Functions
• Fractal algebras of discretization sequences
• Semimartingale Stochastic Approximation Procedures and Recursive Estimation