If $\displaystyle \lim_{n\to\infty}|a_{n+1}/a_n|=L$, then $\displaystyle\lim_{n\to\infty}|a_n|^{1/n}=L$.
There're plenty of proofs available on the internet or books, such as the following one.
There exists an $N$, such that whenever $n>N$,
$$ L-\epsilon<|a_n/a_{n-1}|<L+\epsilon, $$ then $$ (L-\epsilon)^n\frac{|a_1|}{L-\epsilon}<|a_n|<(L+\epsilon)^n\frac{|a_1|}{L+\epsilon}.$$
But I'm doing a problem that specifically says that taking $\log$ and using if $\{a_n\}$ is a sequence converging to $0$, and $\displaystyle s_n=\frac{a_1+a_2+\dots+a_n}{n}$, then $\lim_{n\to\infty}s_n=0$ to prove the statement.
I don't know how to use the hint. Any help?