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Now, if m is odd, then (m + 1 — 2)/2 is an integer which may be taken as the value of b since it satisfies the conditions (7). I t is easily seen that the set (6), in which b = (m + l — 2~)/2 and c is determined by (8), is not the set (4). However, if m is even and not equal to 2~y then b = (m + 2 — 2~)/2 and c determined by (8) are integers which satisfy (7) and yield a set (6) which is not the set (4). But if m = 2~, then N^6 and b = 2 satisfies the conditions (7) on b and yields a set (6) which is not the set (4). An interesting choice of integers b and c is that given by & = ( w + l 2 ^ 4 ) / 2 i f m i s o d d a n d l e s s than or equal to (2-* + 2--l), but by b = (m — 2~*)/2 if m is even and less than or equal to (2~-\-2~). Then (6) require no rearrangement, and c is respectively b or b+l. The resulting integers (6) differ from (4) when m7±2 -*+2-*-l, 2-*+2~*.

Hausdorff methods of summation which include all of the Cesàro methods