An Approximation Inequality for Continued Radicals and Power Forms

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.