# Thread: Series - limits of two series

1. ## Series - limits of two sequences

I have a question in the course infinitesimal calculus:

$\displaystyle a_n , b_n$ are two positive series,
$\displaystyle lim((a_n)/(b_n))=L<infinity$

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

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

Can you please help me with this?

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:

$\displaystyle a_n , b_n$ are two positive series,
$\displaystyle lim((a_n)/(b_n))=L<infinity$

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

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

Can you please help me with this?

It is true: $\displaystyle \lim_{n\to\infty}a_n=\lim_{n\to\infty}b_n\cdot \frac{a_n}{b_n}$ ...now use arithmetic of limits since both sequences converge to a finite limit.

Tonio

4. is it 'legal' to presume that $\displaystyle a_n$ has a finite limit, and use arithmetic of limits?