Limit Comparison Test: Does L Approaching Infinity Matter?

[tmt1]

The limit comparison test states that if $a_n$ and $b_n$ are both positive and $L = \lim_{{n}\to{\infty} } \frac{a_n}{b_n} > 0$ then $\sum_{}^{} a_n$ will converge if $\sum_{}^{} b_n$ and $\sum_{}^{} a_n$ will diverge if $\sum_{}^{} b_n$ diverges. Does this rule also apply if $L$ diverges to infinity?

 

[alyafey22]

Take an example for $a_n,b_n$ and see if the rule satisfies.

 

[tmt1]

Take an example for $a_n,b_n$ and see if the rule satisfies.

It seems to be true based on the examples I've tried, but I'm not sure if I've tried enough examples.

 

[Evgeny.Makarov]

If $a_n>0$ and $b_n>0$, $\lim_{n\to\infty} \frac{a_n}{b_n}=\infty$ and $\sum b_n$ diverges, then $\sum a_n$ also diverges. But for the case when $\sum b_n$ is convergent, take $a_n=1$ and $b_n=1/n^2$.