Convergence Analysis of the Resummation of a Class of Superfactorially Divergent Stieltjes Series by Weniger’s Transformation

The resummation of superfactorially divergent series represents a significant computational challenge in mathematical physics. In the present paper the resummation of a specific class of Stieltjes series characterized by a moment sequence growing as $(2n)!$ will be addressed. Despite the fact that Carleman's condition is satisfied for these series, the practical utility of Padé approximants in resumming them is severely compromised by an extremely slow convergence rate. Weniger's $\delta$-transformation is proposed as a superior resummation tool. Some recently found results on the converging factors of superfactorially divergent Stieltjes series are here used to derive an exact integral representation of the $\delta$ truncation error, which allows for a formal proof of convergence and an analytical asymptotic estimate of the corresponding convergence rate. Numerical experiments are carried out to validate our theoretical findings, confirming that the $\delta$ transformation offers a robust and computationally efficient framework for decoding this class of wildly divergent expansions.