Asymptotic expansions of truncated hypergeometric series for $1\mathrm{/}\pi$

In this paper, we consider rational hypergeometric series of the form $$\frac{p}{\pi }=\sum _{k=0}^{\mathrm{\infty }}{u}_k\text{with}{u}_k=\frac{{\left(\frac{1}{2}\right)}_k{\left(q\right)}_k{(1-q)}_k}{(k!{)}^3}(r+sk)t^k,$$ 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 $${\mathcal{R}}_n=\frac{p}{\pi }-\sum _{k=0}^n{u}_k=\sum _{k=n+1}^{\mathrm{\infty }}{u}_k\mathrm{.}$$ We consider an asymptotic expansion of ${\mathcal{R}}_n$. More precisely, we provide a recursive relation for determining the coefficients $c_j$ such that $${\mathcal{R}}_n=\frac{{\left(\frac{1}{2}\right)}_n{\left(q\right)}_n{(1-q)}_n}{n{!}^3}n{t}^n(\sum _{j=0}^{J-1}\frac{c_j}{n^j}+\mathcal{O}\left(n^{-J}\right)),n\to \mathrm{\infty }\mathrm{.}$$ Here we need $J<\mathrm{\infty }$ to approximate ${\mathcal{R}}_n$, because (like the Stirling series) this series diverges if $J\to \mathrm{\infty }$. By applying our recursive relation to the Chudnovsky formula, we solve an open problem posed by Han and Chen.