Asymptotic expansions for the truncation error in Ramanujan-type series

TL;DR

This paper derives precise asymptotic expansions for the truncation errors in Ramanujan-type series used to compute π, providing detailed error bounds and extending the analysis to all known rational hypergeometric series for 1/π.

Contribution

It introduces asymptotic expansions for the truncation errors in Ramanujan-like series for π, including explicit formulas and bounds, enhancing the understanding of their convergence properties.

Findings

Asymptotic expansion for Chudnovsky series error derived

Explicit error bounds established for finite approximations of π

Extension of asymptotic analysis to all known rational hypergeometric series for 1/π

Abstract

Many of the fastest known algorithms to compute $\pi$ involve generalized hypergeometric series, such as the Ramanujan-Sato series. In this paper, we investigate the rates of convergence for several such series and we give asymptotic expansions for the error of finite approximation. For example, when using the first $n$ terms of the Chudnovskys' series, we obtain the finite approximation $\pi_n\approx \pi$. It is known that the truncation error satisfies $|\pi_n-\pi|\approx 53360^{-3n}.$ In this paper, we prove that the asymptotic expansion for the truncation error in the Chudnovskys' series is $$\left|\pi_n-\pi\right|=53360^{-3n}\cdot\frac{{106720}\sqrt{{10005}\pi}}{{1672209}\sqrt{n}}\cdot\exp\left(\frac{A_1}{n}+\frac{A_2}{n^2}+\frac{\delta_n}{n^3}\right),$$ with ${0.006907}<\delta_n<{0.008429}$ and the exact rational values of $A_1$ and $A_2$: $$A_1= -\frac{1781843197433}{7456754505816},$$ $$A_2= -\frac{1080096011925710088395}{3475199235000451148614116}.$$ Thus we demonstrate how to establish precise error bounds for the approximations for $\pi$ obtained through Ramanujan-like series for $1/\pi$. We also give asymptotic expansions for all known rational hypergeometric series for $1/\pi$ in the appendix.