Some doubt on converges and divergences series

Question

One of tool for determinate the behaviours of the series is to check if $$\limsup_{n\to\infty} \sqrt[n]{a_n}<1$$ or $$\liminf_{n\to\infty} \sqrt[n]{a_n}>1$$ But when the sequences $a_n$ converges to some $l$, the limit is not equal to the $\liminf$ and $\limsup$? So it's wrong determinate directly the $\lim \sqrt[n]{a_n}$? I don't understand why we must check the $\lim \sup$ and $\lim \inf$ and not $\lim \sqrt[n]{a_n}$

Answer 1

There is no reason that $\lim \sqrt[n]{a_n}$ needs to exist. As an example, one could take $a_n = \frac{1}{2^n}$ if $n$ is even and $a_n = 0$ if $n$ is odd. Then $\sqrt[n]{a_n} = \frac 12$ if $n$ is even and $0$ if $n$ is odd, so $\lim \sqrt[n]{a_n}$ doesn't exist, but $\limsup \sqrt[n]{a_n} = \frac 12$.