  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.
