Asymptotic expansions for the truncation error in Ramanujan-type series

Abstract

Many of the fastest known algorithms to compute $π$ 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 $π_n\approx π$. It is known that the truncation error satisfies $|π_n-π|\approx 53360^{-3n}.$ In this paper, we prove that the asymptotic expansion for the truncation error in the Chudnovskys' series is $$\left|π_n-π\right|=53360^{-3n}\cdot\frac{{106720}\sqrt{{10005}π}}{{1672209}\sqrt{n}}\cdot\exp\left(\frac{A_1}{n}+\frac{A_2}{n^2}+\frac{δ_n}{n^3}\right),$$ with ${0.006907}<δ_n<{0.008429}$ 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 $π$ obtained through Ramanujan-like series for $1/π$. We also give asymptotic expansions for all known rational hypergeometric series for $1/π$ in the appendix.