Equivalence of methods for evaluation of sequences
Ralph Palmer Agnew · Proceedings of the American Mathematical Society · 1952
For expositions of the subject, see Hardy [4] and Cooke [3]; our terminology agrees with that of Hardy. Unfortunately, there are no such simple necessary and sufficient conditions that (1.1) be included by convergence, that is, be such that lim Sn = lim an whenever lim an exists. Some Mercerian theorems, which are treated briefly by Hardy [4], solve the problem for special classes of transformations involving a parameter in a simple way, but the general problem never has been and perhaps cannot be successfully attacked. It is the object of this note to prove the following theorem which gives simple sufficient conditions that (1.1) be included by convergence, and to show that the conclusion will fail to follow if the condition (1.31) or any one of the three conditions for regularity is removed from the hypothesis.