Generalization of Ramanujan Method of Approximating root of an equation

TL;DR

This paper generalizes Ramanujan's root approximation method, providing analytical convergence proofs and exploring iterative approaches with arbitrary convergence order.

Contribution

It introduces a generalized framework for Ramanujan's method, including convergence analysis and iterative schemes with customizable convergence rates.

Findings

Convergence of the generalized method is analytically established.

Iterative approaches achieve arbitrary order of convergence.

The method improves root approximation efficiency.

Abstract

We generalize Ramanujan method of approximating the smallest root of an equation which is found in Ramanujan Note books, Part-I. We provide simple analytical proof to study convergence of this method. Moreover, we study iterative approach of this method on approximating a root with arbitrary order of convergence.