Find any sequence that meets these criteria.

Question

I'm struggling with this problem and don't know where to start looking:

Is there any sequence $a_n$ such that $\lim\limits_{n \to \infty}a_n \neq 0$ and $\lim\limits_{n \to \infty}(n \sqrt[n]{|a_n|}) \neq \infty$?

If so, which one? If not, why not?

Can anyone help me out?

EDIT: Also, $\lim\limits_{n \to \infty}a_n \neq \infty$.

Answer 1

Answer to the edited question: suppose that $a_n \to L \not= 0$. Then $|a_n| \to |L|$, but since $|L|^{1/n} \to 1$ you get $\sqrt[n]{|a_n|} \to 1$, so $n\sqrt[n]{|a_n|} \to \infty$.

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How about $0,1,0,1,0,1,\ldots$?

Answer 2

Umberto has a good answer for the case where $a_n$ is not taken to converge. If the second limit does converge, say $$\lim_{n\to\infty}( n\sqrt[n]{|a_n|}) = C$$ then $\sqrt[n]{|a_n|}$ is approximately $C/n$ for large $n$.