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Given $(a_n)$ consider $(s_n)$ given by $s_n = \frac{1}{n}(a_1+a_2+\cdots+a_n)$ for $n \in \Bbb N$. Show that $a_n \rightarrow a$ implies $s_n \rightarrow s$. Further find a divergent sequence $(a_n)$ for which $s_n$ converges.

I've been told to use the Stolz-Cesàro theorem to prove this. However, I'm new to analysis so I couldn't figure out the proof. Is there a more basic way?

Watson
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  • See http://math.stackexchange.com/questions/207910/prove-convergence-of-the-sequence-z-1z-2-cdots-z-n-n-of-cesaro-means?s=8|0.0000 – Watson Sep 08 '16 at 09:59
  • For the counter-example, think about $a_n := (-1)^n$. – Watson Sep 08 '16 at 10:01

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