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Revisiting an infinitely nested radical

Aung Phone Maw

TL;DR

The paper revisits Ramanujan's infinite nested radical problem and develops a formal iteration framework built on three algebraic identities to generate nested radicals. By exploiting these identities, it constructs explicit nested-radical expressions and shows that suitable choices of coefficient sequences $a_i$ and $n_i$ reproduce known results such as $3$, $9$, and $1$, while also yielding new infinite-radical identities. The approach extends Ramanujan's method into a systematic toolkit for closed forms of infinite radicals, linking classical algebraic tricks to iterative representations. The results provide a structured path for discovering additional infinite-radical identities and illustrate the potential for uncovering further relations.

Abstract

We revisit an infinitely nested radical by Ramanujan. Utilizing the full strength of his method, we shall arrive at some new infinitely nested radicals.

Revisiting an infinitely nested radical

TL;DR

The paper revisits Ramanujan's infinite nested radical problem and develops a formal iteration framework built on three algebraic identities to generate nested radicals. By exploiting these identities, it constructs explicit nested-radical expressions and shows that suitable choices of coefficient sequences and reproduce known results such as , , and , while also yielding new infinite-radical identities. The approach extends Ramanujan's method into a systematic toolkit for closed forms of infinite radicals, linking classical algebraic tricks to iterative representations. The results provide a structured path for discovering additional infinite-radical identities and illustrate the potential for uncovering further relations.

Abstract

We revisit an infinitely nested radical by Ramanujan. Utilizing the full strength of his method, we shall arrive at some new infinitely nested radicals.
Paper Structure (2 sections, 7 theorems, 20 equations)

This paper contains 2 sections, 7 theorems, 20 equations.

Table of Contents

  1. Introduction
  2. Three Identities

Key Result

Proposition 1

Let $a_1,a_2,...$ and $n_1,n_2,..$ be two sequences, formally, we have : for all $x$.

Theorems & Definitions (7)

  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Corollary 4
  • Corollary 5
  • Corollary 6
  • Corollary 7