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Let $a_1,a_2,\ldots$ be a sequence of real numbers, and define $s_n=a_1+\ldots+a_n$ for all $n$. Define $t_n=\dfrac{s_1+s_2+\ldots+s_n}{n}$. There is a theorem that if $\{s_n\}$ converges, then $\{t_n\}$ converges (to the same limit, I believe).

Does the converse hold? That is, if $\{t_n\}$ converges, then does $\{s_n\}$ converge?

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  • $\begingroup$ Then this has nothing to do with $a_n$. You can start the question at: "Define $t_n=$... And the answer is no. See Cesaro mean which has the same counterexample as T.Bongers'. $\endgroup$ – Julien Oct 26 '13 at 18:49
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No. Choose $s_n = (-1)^n$; then

$$t_k = \left\{\begin{array}{lr} -\frac{1}{n} &: n \text{ odd} \\ 0 &: n \text{ even} \end{array}\right.$$

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