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Cesaro Summability

• Definition: Let s_n denote the nth partial sum of the series sum(a_n), and let {b_n} be the sequence of arithmetic means defined by b_n = (s_1 + ... + s_n)/n, if n is a natural number. The series sum(a_n) is said to be Cesaro summable (or (C,1) summable) if {b_n} converges. If lim b_n = S, then S is called the Cesaro sum (or (C,1) sum) of sum(a_n).

  Source: Mathematical Analysis, second edition by Tom M. Apostol