If we have a sequence $(a_{n})$ and $(b_{n})$ defined as
$b_{n}=\frac{a_{1}+\dots+a_{n}}{n}$
1) If $a_{n}$ is convergent, is $b_{n}$ convergent?
2) If $b_{n}$ is convergent, is $a_{n}$ convergent?
For the first question, I will use Stolz criterion, setting $c_{n}=n$ and hence, and letting $l$ to be the limit of ${a_n}$
$\frac{a_{n}-a_{n-1}}{c_{n}-c{n-1}}=\frac{a_{n}}{1}=l$
Hence $b_{n}=\frac{a_{n}}{c_{n}}=l$ and $b_{n}$ is convergent.
For the second question, I don't know even how to start. I want to make sure too that the solution of the first question is correct. Any help please?
Let $a_n=(-1)^{n}$. Then $\{b_n\}$ is convergent (t0 $0$) but $\{a_n\}$ is not.