Limit of sequences involving the $n$-th root

Question

Consider a sequence $u_n$ of positive terms. In order to compute the limit of $\sqrt[n]{u_n}$, I know that if $$\dfrac{u_{n+1}}{u_n}\rightarrow a,$$ then also $\sqrt[n]{u_n}\rightarrow a$. Is this still true if $a=+\infty$? Thank you very much!

Answer 1

Yes. Fix $M>0$. If $u_{n+1}/u_n\to\infty$, then there exists $n_0$ such that $u_{n+1}>M\,u_n$ for all $n>n_0$. Thus $$ u_{n_0+k}>M^k u_{n_0}, $$ and then $$\sqrt[n_0+k]{u_{n_0+k}}>M^{k/(n_0+k)}\,u_{n_0}^{1/(n_0+k)}. $$ Taking $k\to\infty$, we get $$ \sqrt[n_0+k]{u_{n_0+k}}>M. $$ As we can do this for each $M>0$, we get that $\sqrt[n]{u_n}\to\infty$.