If $lim_{n\rightarrow\infty} \frac{a_n}{b_n} = c$ and $c>0$, then ${a_n}$ and ${b_n}$ have the same convergence behavior (i.e. if one converges the other converges and if one diverges then the other also diverges).
But what if $c$ is less than $0$? Does that mean no conclusions can be drawn? And what if $c$ is equal to zero?
Any help is appreciated.
If c = 0 then no conclusions can be drawn. We can demonstrate this by providing some counter examples that show the individual sequence can either converge or diverge while the ratio still converges to 0.
For example, take the numerator to be the constant sequence 0 and the denominator to be the constant sequence 1. In this case both converge individually. If you let the numerator alternate between 1 and -1, while the denominator is the sequence 1,2,3,4.... then both will individually not converge.
If c<0 then everything holds just as c>0. This follows from multiplying both sides by -1
The more general then limit test: if $b_n$ absolutely converges, and $\frac{a_n}{b_n}$ is bounded, then $a_n$ also absolutely converges. If $\frac{a_n}{b_n}$ is separated from $0$ and $b_n$ doesn't converge absolutely, then $a_n$ doesn't converge absolutely.
In particular, if there is non-zero limit of $\frac{a_n}{b_n}$, then either both of them converges absolutely, or none of them. If $\frac{a_n}{b_n} \to 0$ and $b_n$ converges absolutely, then $a_n$ converges absolutely.
Nothing can be said about conditional convergence.
