This paper is a research manuscript that I wrote in undergrad.
The goal of this paper is to derive a simple recursion that generates a sequence of fractions approximating √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 sys- tem. Convergence behavior is then discussed in the language of initial trajectories and eigenvectors, effectively proving convergence without notions from standard analysis of infinitesimals.
A natural question at this point would be “Does there exist a structurally simple recursion similar to (3) generating a sequence of fractions approxi- mating the nth root of k ?"