Rogers-Ramanujan continued fraction and approximations to $\mathbf{2\pi}$

TL;DR

This paper explores how the Rogers-Ramanujan continued fraction can be used to generate rapidly converging approximations to 2π, leveraging modular equations for recursive and iterative methods.

Contribution

It introduces new approximation techniques for 2π using Rogers-Ramanujan continued fractions and modular equations, achieving faster convergence than previous methods.

Findings

Approximate 2π with high precision using Rogers-Ramanujan continued fractions.

Recursive formulas based on Ramanujan's modular equations improve accuracy exponentially.

Iterative methods based on Rogers' modular equations converge even faster to 2π.

Abstract

We observe that certain famous evaluations of the Rogers-Ramanujan continued fraction $R(q)$ are close to $2\pi-6$ and $(2\pi-6)/2\pi$, and that $2\pi-6$ can be expressed by a Rogers-Ramanujan continued fraction in which $q$ is very nearly equal to $R^5(e^{-2\pi})$. The value of $-{5\over \alpha}\ln R(e^{-2\alpha \pi})$ converges to $2\pi$ as $\alpha$ increases. For $\alpha=5^n$, a modular equation by Ramanujan provides recursive closed-form expressions that approximate the value of $2\pi$, the number of correct digits increasing by a factor of five each time $n$ increases by one. If we forgo closed-form expressions, a modular equation by Rogers allows numerical iterations that converge still faster to $2\pi$, each iteration increasing the number of correct digits by a multiple of eleven.