Theδ2-process and related topics
Richard R. Tucker · Pacific Journal of Mathematics · 1967
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 3 2 -process are generalized.Some typical results are as follows:(1) If a complex series and its ^-transform converge, their sums are equal.(2) Suppose that Σa n , Σb n are complex series such that Klein -> 0, and A, B exists such that | α n /α Λ -i | ^ A < 1/2, I bjbn-i I £ B < 1 for all sufficiently large n.Then Σb n converges more rapidly than Σa n .(3) If the sequence {l/α Λ -l/α n -i} is bounded, then the complex series Σa n diverges.