An Approximation Inequality for Continued Radicals and Power Forms

TL;DR

This paper develops a generalized approximation inequality for continued radicals and power forms, aiding in convergence analysis and rate estimation, including for Ramanujan's radical.

Contribution

It extends Herschfeld's inequality to arbitrary radicals and power forms, providing new tools for convergence analysis.

Findings

Derived a generalized inequality for continued radicals and power forms.

Applied the inequality to estimate convergence rates of various radicals.

Provided new bounds for Ramanujan's radical.

Abstract

In this article we derive an approximation inequality for continued radicals, generalizing an inequality of Herschfeld for continued square roots to arbitrary radicals, which is useful in exploring convergence issues and obtaining convergence rates. In fact, we generalize this inequality further to encompass the more general continued power forms. We demonstrate the use of this inequality by obtaining estimates for the convergence rates of several continued radicals including the famous Ramanujan radical.