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.

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

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.

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