problem of numeric sequence Give an example of a sequence $(A_n)$ which is not convergent but the sequence $(B_n)$ defined by $\displaystyle B_n = \frac{A_1+A_2+\cdots+A_n}{n}$ is convergent.

You can also consider the alternating $0,1$ sequence i.e. 
$$A_n = 0 \mbox{ if $n$ is odd} $$
$$A_n =1 \mbox{ if $n$ is even }$$
Then $B_n \to 1/2$ but as $A_n$ is oscillatory, it does not converge. For more details, refer Cesaro sums and Cesaro means.
