If $\frac{a_{n+1}}{a_n} \leq \frac{b_{n+1}}{b_n}$ for all $n$, prove series properties $a_n$ and $b_n$ are sequences of non-negative terms
$\dfrac{a_{n+1}}{a_n} \leq \dfrac{b_{n+1}}{b_n}$ for all $n$
If $\sum\limits_{n=1}^\infty b_n$ converges, prove that $\sum\limits_{n=1}^\infty a_n$ converges
If $\sum\limits_{n=1}^\infty a_n$ diverges, prove that $\sum\limits_{n=1}^\infty b_n$ diverges
I was thinking of trying the ratio test.  For the 1st part, $\sum b_n$ converges, which implies that $\lim\limits_{n\to\infty}\left(\dfrac{b_{n+1}}{b_n}\right)$ is less than or equal to $1$.  If it is less than $1$, than the same limit for $\frac{a_{n+1}}{a_n}$ is less than one, which implies $\sum\limits_{n=1}^\infty a_n$ converges.  But I am not sure what to do about if the limit equals $1$.
Thanks for the help

You have the ratio test backwards. It says if $b_{n+1}/b_n \to L <1,$ then $\sum b_n$ converges. It does not say that if $\sum b_n$ converges, then something or other is true about the sequence $b_{n+1}/b_n.$
Note that we need to assume all terms are positive, otherwise there is a problem with the ratios.
Hint:
$$a_n = \frac{a_n}{a_{n-1}}\frac{a_{n-1}}{a_{n-2}} \cdots \frac{a_2}{a_1}a_1.$$
The same is of course true for the $b_n$'s.
