Concerning the ration test for series, I need to prove that $\liminf_{n\to\infty}q_n>1$, with $q_n$ being the quotient sequence, implies divergence. We've already proven that $q_n\geq1$ for almost all $n\in\mathbb{N}$ implies divergence, so could you day that by $\liminf{n\to\infty}q_n>1$, $q_n>1$ for all $n\in\mathbb{N}$ and would this not straightly imply $q_n\geq 1$, i.e. divergence. Why would I then need the strictly greater?
Also, what would be a counter example to the fact that $\limsup_{n\to\infty}q_n>1$ does not suffice for this.