Assume that the following sequences are positive, then the statements:
If $\lim \inf a_{n} > 1$, then $\sum_{k = 1}^{\infty} a_{k}$ diverges.
If $\lim \sup a_{n} = 0$, then $\sum_{k = 1}^{\infty} a_{k}$ converges.
Are both true, correct? However I'm just having trouble visualizing it. What exactly are these statements saying? And what's an example of each? I really just want to be able to see it in a picture because I have a hard time conceptualizing the idea of $\lim \inf$ and $\lim \sup$.
If $\lim \inf a_n > 1$, $a_n$ cannot converge to $0$. So the series diverges and the first statement is true.
For $a_n =\frac{1}{n+1}$, we have $\lim \sup a_n = 0$. As for a sequence converging to $l$, $\lim \inf = \lim \sup = l$. However, $\sum a_n$ diverges. So the second statement is false.