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

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 $-\frac{5}{\alpha }\ln R(e^{-2\alpha \pi })$ 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.