It is well known that there are bounded sequences with divergent Cesàro mean, i.e., a bounded $a_n$ for which $$\lim_{N\to\infty} c_N = \frac{1}{N}\sum_{n=1}^N a_n$$ does not exist. A simple example is $a_n = (-1)^{\lfloor \log_2 n\rfloor}$, for which the $(2^{n+1}-1)$-th term of the sequence of means is $-1/3$ for odd $n$, and the $(2^{n+1}-2)$-th is $1/3$ for even $n$.

But I don't know of any example which is substantially different.

My question is, then, can the Cesàro mean of a bounded sequence diverge in some other way or this slow but stubborn oscillation is essentially the only example?

EDIT: Is there a simple characterization of the bounded sequences for which the Cesàro means converges?

veryclose to 0). – Anthony Quas Jan 2 '12 at 23:25A(containing the terms of the sequence) and a lower triangular matrixCcontaining the coefficients for "implementing" the Cesaro-summation such that $\small C \cdot A = S $ andScontains a sequence approaching the convergent. Then it seems simple to invent any non-converging vectorS(say, having periodic entries) and premultiply with the inverse ofC: $\small C^{-1} \cdot S = B $ which produces then inBan interesting(?) sequence. – Gottfried Helms Jan 3 '12 at 0:15Divergent Seriespublished in 1948 (which I haven't got, unfortunately). This old-fashioned book, by one of the last surviving champions of "hard analysis", is unfortunately not very well-known nowadays, probably because "soft analysis", e.g. functional analysis was much more fashionable then - but renewed versions of "hard analysis" are making comebacks! – Zen Harper Jan 3 '12 at 3:02Tauberian theoremswhich say that if the Cesaro means (or other means) convergeandthe sequence $(a_n)$ satisfies an extra condition (called aTauberian condition) then the sequence itself converges (necessarily to the same limit). Famous Tauberian conditions include Littlewood's $na_n > -K$ condition, and stuff aboutslowly oscillatingsequences, and many, many other conditions also; as well as stuff used to prove the Prime Number Theorem. As well as Hardy's book, I recommendTauberian theory, a century of developments, by Jacob Korevaar. – Zen Harper Jan 3 '12 at 3:15