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

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.

On the Euler sequence spaces which include the spaces l p and l ∞ I

Convergence acceleration during the 20th century

It is known that this method of summability cannot be included by any matrix method, but for bounded sequences it is included by the Cesáro matrix method C\\. In this paper these results eure extended by comparing statistical convergence with the intersection of a collection Τ of summability matrices, each of which is somewhat like C\\. It is shown that a bounded sequence is statistically convergent if and only if it is summable by every matrix in T . On the other hand, no countable collection of matrices can include statistical convergence for unbounded sequences. Also, the class Τ is studied to determine which classical summability matrices belong to T .

Robert Ellis Powell

Papers

Summability theory and its applications