Nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series

TL;DR

This paper discusses various nonlinear sequence transformations, including new ones, for accelerating convergence and summing divergent series, with theoretical analysis and practical testing on scientific series.

Contribution

Introduces new nonlinear sequence transformations and provides efficient algorithms, along with theoretical analysis and performance testing on scientific series.

Findings

Effective acceleration of convergence demonstrated

New transformations outperform traditional methods

Transformations successfully sum divergent series

Abstract

Slowly convergent series and sequences as well as divergent series occur quite frequently in the mathematical treatment of scientific problems. In this report, a large number of mainly nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series are discussed. Some of the sequence transformations of this report as for instance Wynn's $\epsilon$ algorithm or Levin's sequence transformation are well established in the literature on convergence acceleration, but the majority of them is new. Efficient algorithms for the evaluation of these transformations are derived. The theoretical properties of the sequence transformations in convergence acceleration and summation processes are analyzed. Finally, the performance of the sequence transformations of this report are tested by applying them to certain slowly convergent and divergent series, which are hopefully realistic models for a large part of the slowly convergent or divergent series that can occur in scientific problems and in applied mathematics.