Suppose $\sum_{n>1} a_n=\infty$ and $0<a_{n+1}<a_n$.
Let $b_k=a_k$ or $b_k=0$ for all integers $k$.
Let $R=\lim_{n\rightarrow\infty}((1/n)\sum_{q=1}^{q=n} b_q/a_q)$
If $R>0$, how to show that $\sum_{n>1}b_n=\infty$?
If $0<\lim_{n\rightarrow\infty}((1/\sqrt n)\sum_{q=1}^{q=n} b_q/a_q)$, must $\sum_{n>1}b_n=\infty$?
Per OP's request, posting the above comment as an answer.
I believe the first part of your question is answered by Theorem 2 in the paper Tibor Šalát: On subseries, Mathematische Zeitschrift, Volume 85, Number 3, 209-225.
For the second part $a_n=\frac1n$ and
$b_n= \begin{cases} a_n,& n=k^2\\ 0,& \text{otherwise} \end{cases}$
should work as a counterexample.