Problem 203 (difficulty: 4/10)

Consider the sequence \(\displaystyle s_n\) of arithmetic means

\(\displaystyle s_n=\frac{a_1+\ldots +a_n}{n} \)

corresponding to the sequence \(\displaystyle a_n\). Show that if \(\displaystyle \lim\limits_{n\to \infty} a_n=a,\) then \(\displaystyle \lim\limits_{n\to\infty} s_n =a.\) Give an example when \(\displaystyle (s_n)\) is convergent, but \(\displaystyle (a_n)\) is divergent.

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