  In this paper, we consider rational hypergeometric series of the form
\[\frac{p}{\pi}= \sum_{k=0}^\infty u_k\quad\text{with}\quad
u_k=\frac{\left(\frac{1}{2}\right)_k \left(q\right)_k
\left(1-q\right)_k}{(k!)^3}(r+s\,k)\,t^k,\] where $(a)_k$ denotes the
Pochhammer symbol and $p,q,r,s,t$ are algebraic coefficients. Using only the
first $n+1$ terms of this series, we define the remainder \[\mathcal{R}_n =
\frac{p}{\pi} - \sum_{k=0}^n u_k=\sum_{k=n+1}^\infty u_k.\] We consider an
asymptotic expansion of $\mathcal{R}_n$. More precisely, we provide a recursive
relation for determining the coefficients $c_j$ such that \[ \mathcal{R}_n =
\frac{\left(\frac{1}{2}\right)_n \left(q\right)_n
\left(1-q\right)_n}{n!^3}nt^n\left(\sum_{j=0}^{J-1}\frac{c_j}{n^j}+\mathcal{O}\left(n^{-J}\right)\right),\qquad
n \rightarrow \infty.\] Here we need $J<\infty$ to approximate $\mathcal{R}_n$,
because (like the Stirling series) this series diverges if
$J\rightarrow\infty$. By applying our recursive relation to the Chudnovsky
formula, we solve an open problem posed by Han and Chen.
