Inclusion relations between power methods of limitation

Let (Formula Presented) be a power series with pk(k = 0, 1,.) complex numbers and 0 <ρp≦ ∞ its radius of convergence, and assume that P(x) ≠ 0 for 0 ≦ αp ≦ x < ρp. The powermethod of limitation, P, is defined by (Formula Presented) (provided the series converges in [αp, ρp) and the limit existsand is finite). Abel and Borel methods are the best known power methods. In this article inclusion relations between two power methods are investigated. Several theorems areproved, which lead to necessary and sufficient conditions, forinclusion, that are correct under some fairly moderate restrictions. © 1976, University of California, Berkeley. All Rights Reserved.