Convergence Analysis of the Summation of the Euler Series by Pad\'e Approximants and the Delta Transformation

Sequence transformations are valuable numerical tools that have been used with considerable success for the acceleration of convergence and the summation of diverging series. However, our understanding of their theoretical properties is far from satisfactory. The Euler series $\mathcal{E}(z) \sim \sum_{n=0}^{\infty} (-1)^n n! z^n$ is a very important model for the ubiquitous factorially divergent perturbation expansions in physics. In this article, we analyze the summation of the Euler series by Pad\'e approximants and the delta transformation [E. J. Weniger, Comput. Phys. Rep. Vol.10, 189 (1989), Eq. (8.4-4)] which is a powerful nonlinear Levin-type transformation that works very well in the case of strictly alternating convergent or divergent series. Our analysis is based on a new factorial series representation of the truncation error of the Euler series [R. Borghi, Appl. Num. Math. Vol.60, 1242 (2010)]. We derive explicit expressions for the transformation errors of Pad\'e approximants and of the delta transformation. A subsequent asymptotic analysis proves \emph{rigorously} the convergence of both Pad\'e and delta. Our asymptotic estimates clearly show the superiority of the delta transformation over Pad\'e. This is in agreement with previous numerical results.