Lim sup definition doesn't make sense to me

Question

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!

Answer 1

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:

$$ \begin{align*} \limsup_{n\to\infty} s_n &= 5 \\ \liminf_{n\to\infty} s_n&= 0\end{align*}$$

Answer 2

Let $\{a_n\}$ be any sequence. Look at the supremum of the sequence starting at the first term, and then at the second, and so on: \begin{align*} c_1 &= \sup \{a_1, a_2, a_3,\ldots\} \\ c_2 &= \sup \{a_2, a_3, a_4, \ldots\} \\ c_3 &= \sup \{a_3, a_4, a_5, \ldots\} \end{align*} and so on. If $c_1 = \infty$ then in fact $c_n = \infty$ for all $n$ so that $c_n \to \infty$. On the other hand, if $c_1 < \infty$ we have $c_1 \ge c_2 \ge c_3 \ge \cdots$. Since any nonincreasing sequence converges (possibly to $-\infty$) the sequence $\{c_n\}$ converges. Thus $\lim_{n \to \infty} c_n$ makes sense regardless of whether or not $\{a_n\}$ converges. The $\limsup$ of $\{a_n\}$ is just this limit.

Answer 3

Consider the sequence

$$ 0, 1.1, 0, 1.01, 0, 1.001, 0, 1.0001, 0, 1.00001, 0, \ldots $$

This series is clearly divergent, but there is still information content in its limiting behavior: e.g. it has two limit points, $0$ and $1$. The lim sup of this sequence is $1$, and the lim inf is 0.

I pick this example, because it demonstrates we need something more sophisticated than simply the supremum of the sequence, which is $1.1$. The lim sup can be interpreted as the limiting behavior of "the supremum of the tail".