The δ²-process and related topics

Richard R. Tucker

Vol. 22 (1967), No. 2, 349–359

DOI: 10.2140/pjm.1967.22.349

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 Σa_n,Σb_n are complex series such that h_n∕a_n → 0, and A,B exists such that |a_n∕a_n−1|≦ A < 1∕2, |b_n∕b_n−1|≦ B < 1 for all sufficiently large n. Then Σb_n converges more rapidly than Σa_n.

(3) If the sequence {1∕a_n − 1∕a_n−1} is bounded, then the complex series Σa_n diverges.

Mathematical Subject Classification

Primary: 40.13

Milestones

Received: 25 April 1966

Published: 1 August 1967