Iterated Radical Expansions and Convergence

TL;DR

This paper explores three radical recurrences, including a famous infinite radical for the Golden mean, providing new proofs and analyzing their convergence rates and asymptotic behaviors.

Contribution

It introduces a novel proof for a key formula related to the Golden mean radical expansion and studies the convergence properties of two additional non-exponential recurrences.

Findings

New proof of the Golden mean radical expansion formula

Analysis of convergence rates for three radical recurrences

Asymptotic series with intrinsic constants depending on initial conditions

Abstract

We treat three recurrences involving square roots, the first of which arises from an infinite simple radical expansion for the Golden mean, whose precise convergence rate was made famous by Richard Bruce Paris in 1987. A never-before-seen proof of an important formula is given. The other recurrences are non-exponential yet equally interesting. Asymptotic series developed for each of these two examples feature a constant, dependent on the initial condition but otherwise intrinsic to the function at hand.