The δ²-process and related topics. II

Richard R. Tucker

Vol. 28 (1969), No. 2, 455–463

DOI: 10.2140/pjm.1969.28.455

Abstract

This paper considers three transforms of a complex series ∑ a_n: namely, (1) Aitken’s ∂²-transform ∑ b_n, (2) Lubkin’s W-transform ∑ c_n, and (3) a closely related transform ∑ d_n which the author calls the W1-transform and for which ∑ ₀ⁿd_k = ∑ ₀ⁿ⁺¹c_k. If a_n−1≠0, set r_n = a_n∕a_n−1. If, moreover, ∑ a_n converges, define T_n = (a_n + a_n+1 + ⋯)∕a_n−1 and let MR(∑ a_n) be the class of all series converging more rapidly to the sum S = ∑ a_n than ∑ a_n. Some of the results proven in this paper are as follows:

(1) If b_n∕a_n → 0, then the three conditions (i) ∑ b_n ∈ MR(∑ a_n), (ii) ∑ c_n ∈ MR(∑ a_n), and (iii) ∑ d_n ∈ MR(∑ a_n) are equivalent.

(2) ∑ b_n ∈ MR(∑ a_n) if and only if ΔT_n → 0.

(3) If |r_n|≦ ρ < 1 for all sufficiently large n, then the three conditions (i) ∑ b_n ∈ MR(∑ a_n), (ii) Δr_n → 0, and (iii) b_n∕a_n → 0 are equivalent.

Mathematical Subject Classification

Primary: 40.13

Milestones

Received: 27 March 1967

Published: 1 February 1969