<s>[INST] Cesàro sum of $1+ 0 - 1 + 1 + 0 - 1 + \dots$ I am trying to compute the Cesàro sum of $1+ 0 - 1 + 1 + 0 - 1 + \dots$. When I compute the Cesàro means, I get the following sequence
$$\left(1, 1, \frac{2}{3}, \frac{3}{4}, \frac{4}{5}, \frac{4}{6}, \frac{5}{7}, \frac{6}{8}, \frac{6}{9}, \frac{7}{10}, \frac{8}{11}, \frac{8}{12}, \cdots\right)$$
Where does this sequence converge to? Is it $\frac{2}{3}$? I fail to see the pattern of this sequence. If I can just rewrite the sequence into a more general form, then I might be able to compute the limit.
 [/INST] Lets say the first term is the $0$th term. If $s_n,n=0,1,2,3,\dots$ are the partial sums,
$$ s_{3n} = 1,\\s_{3n+1}=1,\\s_{3n+2}=0.$$
Consequently
$$ \sum_{n=0}^{3k} s_n = 2k+1, \\\sum_{n=0}^{3k+1} s_n= 2k+2,  \\\sum_{n=0}^{3k+2}s_n = 2k+2.  $$
The Cesàro sums are the average of the partial sums $c_k = \frac1{k+1}\sum_{n=0}^k s_n$, and its easy to check via squeeze theorem that $c_{3k}\to 2/3$, $c_{3k+1}\to 2/3$, $c_{3k+2}\to 2/3$. Hence $c_{k}\to 2/3$.
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