It is very similar to this question that I posted not a while ago,
But I'm still having a hard time to transalte or use the solution that was given.
Now ,the sequence $a_{_{n}}$ applies these condition:
$a_{_{n}}\geq 0$ for every $n \in \mathbb{N}$
$\lim_{n\to \infty }\sqrt[n]{a_{n}}< 1$
again,I need to prove that $a_{n}$ is convergent, and it's limit is 0.
As was suggested I tried to follow the limit defenition.