I have hard time understanding why the concept of lim superior is necessary, for example the ratio test. There are two ways, it seems of phrasing this:
(Abbott, Understanding Analysis p.78) Given $\sum a_n$ with $a_n \neq 0$, if $\lim \lvert{\frac{a_{n+1}}{a_n}} \rvert = r < 1$, then the series converges absolutely
(Rudin, Principles of Mathematical Analysis p. 66) The series $\sum a_n$ converges if $\lim \sup _{n \to \infty} \lvert{\frac{a_{n+1}}{a_n}} \rvert < 1$
Why is the addition of supremum necessary, if the first one seems just fine? What exactly is the difference between THE limit and the supremum of the set of subsequential limits?
Take the sequence $(a_n)_n$ defined by $a_1=1$ and $$ a_{n+1} = \frac{2+(-1)^n}{4}\cdot a_n $$ Then, obviously, $$ 0 \leq \frac{a_{n+1}}{a_n} = \frac{2+(-1)^n}{4} $$ You can comnpute its limsup and apply the second theorem. The limit, however, doesn't exist, so you cannot apply the first. So the second theorem is strictly more general than the first.
(Note, of course, that if the limit exists, then it is equal to the limsup and both theorems apply.)