Lim sup definition doesn't make sense to me I don't really understand the concept of Lim Sup and why it is any different from a standard limit. I am working on a question about radius of convergence and I know that $r = \frac{1}{\limsup\limits_{n\to\infty} A_n^\frac{1}{n}}$ but don't have any intuitive idea of what this means. Could someone put it into words that I can understand please? Thanks and sorry for my inability to format things correctly on here!

Difference of $\limsup$ and $\lim$
If you have a series that is not convergent like this one:
$$ s_n := \left\{\begin{matrix} \frac{1}{n} & n \text{ is even}\\ 5+\frac{1}{n^2} & n \text{ is not even} \end{matrix}\right.$$
Than your Limes just does not exists, in a lot of cases it is still possible to deduce information from such series. By just taking the "biggest limes" or biggest limit point. 
So whats the $\limsup$ here basically:
$$ \begin{align*} \limsup_{n\to\infty} s_n &= 5 \\ \liminf_{n\to\infty} s_n&= 0\end{align*}$$ 
