Originally Posted by

**roninpro** You should try to find a better teacher!

Suppose that $\displaystyle \lim_{n\to \infty} A_n/B_n=L>0$. This means that the sequence $\displaystyle \{A_n/B_n\}$ can be bounded above and below, say

$\displaystyle m<A_n/B_n<M$

Multiplying both sides by $\displaystyle B_n$: $\displaystyle mB_n<A_n<MB_n$. Then we have

$\displaystyle m\sum B_n<\sum A_n<M \sum B_n$

If $\displaystyle \sum A_n$ converges, then $\displaystyle \sum B_n<(1/m) \sum A_n$, so $\displaystyle \sum B_n$ converges by comparison. On the other hand, if $\displaystyle \sum B_n$ converges, then $\displaystyle \sum A_n<M \sum B_n$, so $\displaystyle \sum A_n$ converges.

In summary, if the limit condition above holds, then $\displaystyle \sum A_n$ and $\displaystyle \sum B_n$ either both converge or diverge.

Let us know if you have any questions about this.