# Thread: Series - limits of two series

1. ## Series - limits of two sequences

I have a question in the course infinitesimal calculus:

$a_n , b_n$ are two positive series,
$lim((a_n)/(b_n))=L

Prove or give a negative example: if $lim(b_n)=0$, then $lim(a_n)=0$.

It looks very tricky, though I just couldn't find a negative example, or a way to prove it.

2. I am not the best at these things, but is that not similar to the limit comparison test? Would the proof of that not help you? just curious

3. Originally Posted by adam63
I have a question in the course infinitesimal calculus:

$a_n , b_n$ are two positive series,
$lim((a_n)/(b_n))=L

Prove or give a negative example: if $lim(b_n)=0$, then $lim(a_n)=0$.

It looks very tricky, though I just couldn't find a negative example, or a way to prove it.

It is true: $\lim_{n\to\infty}a_n=\lim_{n\to\infty}b_n\cdot \frac{a_n}{b_n}$ ...now use arithmetic of limits since both sequences converge to a finite limit.
4. is it 'legal' to presume that $a_n$ has a finite limit, and use arithmetic of limits?