We just had the root test in class:
$\sum_{n=1}^\infty a_n$ (in $\mathbb R)$ converges if $\lim\limits_{n\rightarrow\infty}\sup\sqrt[n]{|a_n|}<1$
Why is it important to take the $\lim\sup$ and not taking just $\lim$? Any examples?
I've considered some series but with none of them I had a problem of taking $\lim$ instead of $\lim\sup$.