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Consider the sequence $(0,0,1,1,1,1,0,0,0,0,0,0,0,0,1,1..)$, i.e the sequence of 2 zeros, followed by $2^2$ ones, followed by $2^3$ zeros, $2^4$ ones and so on. I know that the sequence of Cesaro means of this sequence doesn't converge, but I am not sure how to prove this. I can just see that they increase over the places that have ones and then decrease when the zeros begin.

Also, what about the Cesaro means of the sequence of Cesaro means? Do they converge?

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Strategy:

If you add up the first $2^n$ terms, then:

If $n$ is even then over $\frac{1}{2}+\frac{1}{8}$ of the terms will be $1$, so the sum is over $\frac{5}{8}2^n$.

If $n$ is odd then over $\frac{1}{2}+\frac{1}{8}$ of the terms will be $0$, so the sum is under $\frac{3}{8}2^n$.

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