Determination of $\mathbf{f}(\mathbf{\infty })$ from the Asymptotic Series for $\mathbf{f}(\mathbf{x})$ About $\mathbf{x}=\mathbf{0}$

A difficult and long-standing problem in mathematical physics concerns the determination of the value of $f(\mathrm{\infty })$ from the asymptotic series for $f(x)$ about $x\text{\hspace{0.17em}⁣}=\text{\hspace{0.17em}⁣}0$. In the past the approach has been to convert the asymptotic series to a sequence of Pad'e approximants $\{P_n^n(x)\}$ and then to evaluate these approximants at $x\text{\hspace{0.17em}⁣}=\text{\hspace{0.17em}⁣}\mathrm{\infty }$. Unfortunately, for most physical applications the sequence $\{P_n^n(\mathrm{\infty })\}$ is slowly converging and does not usually give very accurate results. In this paper we report the results of extensive numerical studies for a large class of functions $f(x)$ associated with strong-coupling lattice approximations. We conjecture that for large $n$, $P_n^n(\mathrm{\infty })\text{\hspace{0.17em}⁣}\sim \text{\hspace{0.17em}⁣}f(\mathrm{\infty })+B\mathrm{/}\ln n$. A numerical fit to this asymptotic behavior gives an accurate extrapolation to the value of $f(\mathrm{\infty })$.