A contribution to the theory of divergent sequences

In this paper we define and examine a new method of summation which x This assigns a general limit Lim x,~ to certain bounded sequences x-~ xx~j. method is analogous to the mean values which are used in the theory of almost periodic functions, furthermore it is narrowly connected with the limits of S. BANXCH. a The sequences which are summable by this method F we shall call almost convergent. In spite of the fact that our method contains certain classes of matrix methods (for bounded sequences) it is not strong (w 3). Its most remarkable property is that most of the commonly used matrix methods contain the method F (w 5). In spite of this T' is equivalent to none of the matrix methods (w 7).