Summation of sequences and summation of series
G. G. Lorentz, Karl Zeller · Proceedings of the American Mathematical Society · 1964
of the series Eun, un = SnSn-1, S=0, into the sequence {rn}. Conditions for the regularity of the methods A, A' are well known [1, pp. 64, 68], [2, pp. 389, 396]. If the summability field ? of a sequence or series method A (i.e., the set of all sequences s, summable by A) is contained in the summability field e3 of another method B (i.e., if 2 Ce), we call B stronger than A. It is natural to call the matrix A' of (2) dual to the matrix A of (1) (or A dual to A') if awn becomes rTn (or, respectively, rn becomes an) under the application of the formal summation by parts; this is equivalent to the relation a' = =a(or to the relation aik = aan'+J). In many concrete cases, dual methods of summation are equivalent, in the sense that they define the same summability fields and the same limits. One can also give simple sufficient conditions which guarantee this ([4, Theorems 8, 9], a misprint should be corrected there: A is to be replaced by B). It is easy to give examples of dual methods of summation of opposite types which are not equivalent in this sense, but this does not exclude the possibility that one of them is equivalent to some other method of the opposite type. In this paper we show that sequence and series methods are essentially different: There exist regular sequence summation methods for which the summability field is not contained in the summability field of any regular