I'm confused by one thing
lct states that if you have two series, $a_n$ and $b_n$ if taking the limit of two (where $bn$ is the decent comparison where it's positive for all n meaning it's limit doesn't equal 0) then both series behave the same way, ie both converge or diverge.
If $L = 0$ (which you can only have if $a_n$'s limit is 0 as the limit of $b_n$ can't be zero) and you know $b_n$ converges then $a_n$ must also converge, this I understand due to the vanishing condition, you can only get 0 if $a_n$'s limit is 0.
What is confusing me this, if $L=\infty$ and you know $b_n$ diverges, how does this imply $b_n$ also diverges? You would have an instance of an indeterminate form right where it's $\frac{\infty}{\infty}$.