# Limit comparison test

• MHB
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?

## Answers and Replies

Gold Member
MHB
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.

Gold Member
MHB
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 $b_n=1/n^2$.