Bounded sequence with divergent Cesaro means Is there a bounded real-valued sequence with divergent Cesaro means (i.e. not Cesaro summable)?
More specifically, is there a bounded sequence $\{w_k\}\in l^\infty$ such that $$\lim_{M\rightarrow\infty} \frac{\sum_{k=1}^M w_k}{M}$$ does not exist?
I encountered this problem while studying the limit of average payoffs criterion for ranking payoff sequences in infinitely repeated games.

Consider $1,-1,-1,1,1,1,1,-1,-1,-1,-1,-1,-1,-1,-1,\cdots$ (one $1$, two $-1$, four $1$, eight $-1$, ...) Then 
$$\frac{1-2+2^2-2^3+\cdots+(-2)^n}{1+2+2^2+\cdots+2^n}=\frac{(-2)^{n+1} -1}{3(2^{n+1}-1)}$$
This sequence is divergent. So $(\sum_{k\le M}a_k)/M$ has divergent subsequence, and it implies nonexistence of Cesaro mean of $a_n$.
