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Suppose $\sum_{n>1} a_n=\infty$ and $0.
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 $\sum_{n>1}b_n=\infty$?

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    I believe the first part of your question is answered by Theorem 2 in the paper Tibor Šalát: [On subseries](http://dx.doi.org/10.1007/BF01112142), Mathematische Zeitschrift, Volume 85, Number 3, 209-225.\\For the second part an=1/n and bn=an for n=k2, bn=0 for n≠k2 should work as a counterexample.2011-11-20

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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.

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    The paper is also freely available [a$t$ GDZ](h$t$tp://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN266833020_0085&DMDID=DMDLOG_0029).2012-09-25