2
$\begingroup$

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 $\lim\sup$.

  • 5
    The limsup always exists, unlike ordinary limits. So, in most practical applications you can just use regular $\lim$. Fortunately, even when the limit doesn't exist, you can still get information via the limsup.2012-12-14
  • 0
    Can you please be more specific? What does "limsup does always exist" mean?2017-11-12

3 Answers 3