So I have an exercise where I need to show the following statement and I can't find a possible way to prove it, so help would be appreciated.
The sequence $(a_k)_k$ and $(b_k)_k$ are sequnces of complex or real numbers, with $b_k \neq0$ for all $k \in \mathbb{N} $.
Show: If the sequence $\left(\frac{|a_k|}{|b_k|}\right)_k \ $ is convergent, with $\lim_{k \rightarrow \infty} \frac{a_k}{b_k} > 0 $, then the series $\sum_{k=1}^{\infty}a_k \ $ is absolutely convergent, if and only if the series $\sum_{k=1}^{\infty}b_k$ is absolutely convergent
For complex numbers case, the limit should have with the absolute value, if not, saying $>0$ is strange.
Assume $L=\lim_{k}|a_{k}|/|b_{k}|>0$, then there exists some $N$, for all $k\geq N$, $\left|\dfrac{|a_{k}|}{|b_{k}|}-L\right|<1$, then $\dfrac{|a_{k}|}{|b_{k}|}<L+1$, so $|a_{k}|<(L+1)|b_{k}|$, and if $\displaystyle\sum_{k}|b_{k}|<\infty$ then $\displaystyle\sum_{k}|a_{k}|\leq(L+1)\sum_{k}|b_{k}|<\infty$.