Vol. 22, No. 2, 1967

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ISSN: 0030-8730
The δ2-process and related topics

Richard R. Tucker

Vol. 22 (1967), No. 2, 349–359
Abstract

This paper deals with (1) acceleration of the convergence of a convergent complex series, (2) rapidity of convergence, and (3) sufficient criteria for the divergence of a complex series. Various results of Samuel Lubkin, Imanuel Marx and J. P. King which concern or are closely related to Aitkin’s δ2-process are generalized. Some typical results are as follows:

(1) If a complex series and its δ2-transform converge, their sums are equal.

(2) Suppose that Σan,Σbn are complex series such that hn∕an 0, and A,B exists such that |an∕an1|A < 12, |bn∕bn1|B < 1 for all sufficiently large n. Then Σbn converges more rapidly than Σan.

(3) If the sequence {1∕an 1∕an1} is bounded, then the complex series Σan diverges.

Mathematical Subject Classification
Primary: 40.13
Milestones
Received: 25 April 1966
Published: 1 August 1967
Authors
Richard R. Tucker