The difference with sup and without sup The difference with sup and without sup，how to judge and choose the use

For example, here is rudin's "Root" Test:
Given $\sum a_n$, put $\color{Green}{\{\alpha =\lim_{n\to \infty}\sup \sqrt[n]{\left|a_n\right|}\}}$
Here is mathworld "Root" Test

I also see SupremumLimit
So, as title shows, and the question may not only occurs in this example.
Others maybe RatioTest, ... other Tests, the use of Sup and Inf and without use, when should I use, and when should I not use?
 A: Generally speaking: the texts that use limsup instead of lim are just being more careful.  You can certainly prove that if $\lim_{k\rightarrow\infty}u_k^{1/k}$ exists, and call its value $\rho$, then the root test applies: if $\rho>1$ you get divergence, whereas you get absolute convergence for $\rho<1$.
However, the result given by Rudin is stronger!  Whenever the limit exists, necessarily the limsup also exists and gives the same values; but, there are certainly sequences which do not have a limit but which still have a limit supremum.
A: Rudin's statement is more general and doesn't require you to prove the existence of a limit. This is because the limsup of a (real) sequence is always defined as an extended real number (i.e., as either a real number or $\pm \infty$), while not all sequences have limits. So Rudin's version of the root test doesn't make any assumptions about the sequence $\{\sqrt[n]{|a_n|}\}$, and hence it can be applied to any series $\sum a_n$. In contrast, to apply the version of the root test using $\lim \sqrt[n]{|a_n|}$, you first need to prove that this limit exists.
It's also worth noting that $\lim \sqrt[n]{|a_n|} = \limsup \sqrt[n]{|a_n|}$ whenever $\lim \sqrt[n]{|a_n|}$ exists, and so there is no real benefit to using the $\lim \sqrt[n]{|a_n|}$ version of the root test. (Of course, analogous statements can be made about the ratio test, too.)
