# Dense subseries of divergent series

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 $\sum_{n>1}b_n=\infty$?

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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 an=1/n and bn=an for n=k2, bn=0 for n≠k2 should work as a counterexample. –  Martin Sleziak Nov 20 '11 at 16:24
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## 1 Answer

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 at GDZ. –  Martin Sleziak Sep 25 '12 at 17:08
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