For any sequence $(a_n)$ of positive numbers, we have $\liminf_{n\to\infty}\frac{a_{n+1}}{a_n}\leq\liminf_{n\to\infty}\sqrt[n]{a_n}\leq\limsup_{n\to\infty}\sqrt[n]{a_n}\leq\limsup_{n\to\infty}\frac{a_{n+1}}{a_n}.$
The above chain of inequalities lets us show that if $\cfrac{a_{n+1}}{a_n}$ converges, or diverges to $\infty$, then (respectively) so does $\sqrt[n]{a_n}$--moreover, in the case of convergence, they share the same limit. We can use this approach to show that $(x_n)$ converges and that $(y_n)$ diverges to $\infty$.
Even if $\cfrac{a_{n+1}}{a_n}$ neither converges nor diverges to $\infty$--that is, if $\liminf\limits_{n\to\infty}\cfrac{a_{n+1}}{a_n}<\limsup\limits_{n\to\infty}\cfrac{a_{n+1}}{a_n}$--it is still possible that $\sqrt[n]{a_n}$ can converge or diverge to $\infty$, so this method doesn't work all the time. Still, it's handy to keep it in our toolbox when we're dealing with sequences of the form $(\sqrt[n]{a_n})$.