Vol. 28, No. 2, 1969

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

Richard R. Tucker

Vol. 28 (1969), No. 2, 455–463
Abstract

This paper considers three transforms of a complex series an: namely, (1) Aitken’s 2-transform bn, (2) Lubkin’s W-transform cn, and (3) a closely related transform dn which the author calls the W1-transform and for which 0ndk = 0n+1ck. If an10, set rn = an∕an1. If, moreover, an converges, define Tn = (an + an+1 + )∕an1 and let MR( an) be the class of all series converging more rapidly to the sum S = an than an. Some of the results proven in this paper are as follows:

(1) If bn∕an 0, then the three conditions (i) bn MR( an), (ii) cn MR( an), and (iii) dn MR( an) are equivalent.

(2) bn MR( an) if and only if ΔTn 0.

(3) If |rn|ρ < 1 for all sufficiently large n, then the three conditions (i) bn MR( an), (ii) Δrn 0, and (iii) bn∕an 0 are equivalent.

Mathematical Subject Classification
Primary: 40.13
Milestones
Received: 27 March 1967
Published: 1 February 1969
Authors
Richard R. Tucker