  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.
