an:06440686
Zbl 1325.65008
Borghi, Riccardo; Weniger, Ernst Joachim
Convergence analysis of the summation of the factorially divergent Euler series by Pad?? approximants and the delta transformation
EN
Appl. Numer. Math. 94, 149-178 (2015).
00344606
2015
j
65B10 41A21 40D05 40G99
Euler series; summation of factorial divergence; Weniger's delta transformation; Pad?? approximants; convergence proofs
Summary: 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 theoretical physics and for the divergent asymptotic expansions for special functions. In this article, we analyze the summation of the Euler series by Pad?? approximants and by the delta transformation, 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 very recent factorial series representation of the truncation error of the Euler series. We derive explicit expressions for the transformation errors of Pad?? approximants and of the delta transformation. A subsequent asymptotic analysis proves \textit{rigorously} the convergence of both Pad?? and delta. Our asymptotic estimates clearly show the superiority of the delta transformation over Pad??. This is in agreement with previous numerical results.