Asymptotic expansions for the truncation error in Ramanujan-type series

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.