I want to show that for any convergent series $\sum_{n=0}^{\infty}a_n$ with $a_n>0$ there exists a sequence $(b_n)$, $\lim\limits_{n \to \infty} b_n = \infty$, such that $\sum_{n=0}^{\infty}a_nb_n$ converges. Any hints?
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Look at the ratio test. From the limit of $\frac{a_{n+1}}{{a_n}}$ construct a greater limit < 1 for $\frac{a_{n+1}b_{n+1}}{a_n b_n}$ and from there the sequence $b_n$. – aventurin Dec 05 '16 at 22:58
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@aventurin what if the ratio test fails for $\sum a_n$? – Ben Grossmann Dec 05 '16 at 23:26
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Hint: Define a sequence $(n_k)$ such that for each $k$, $$ \sum_{n=n_k}^\infty a_n < \frac{1}{4^k} $$ use these $n_k$ to define a sequence $(b_n)$ in which for all $k$, $$ b_{n_k} = b_{n_k+1} = \cdots = b_{n_{k+1}- 1} $$
Ben Grossmann
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@TonyK I do not, though I suppose the idea that gets at also works – Ben Grossmann Dec 05 '16 at 22:33
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Then your answer is rather gnomic. You should perhaps make explicit that $(b_n)$ tends to infinity. (I know this is part of the question, but it's not part of your answer.) – TonyK Dec 05 '16 at 22:38
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@TonyK gnomic is a cool word, thanks for that. I appreciate your objection, but I feel that since the OP is looking for a hint on how to approach the problem, this is an appropriate answer. – Ben Grossmann Dec 05 '16 at 22:44
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Hint: There are positive integers $n_1<n_2 < \cdots$ such that
$$\sum_{n=n_k}^{n_{k+1}-1} a_n < \frac{1}{2^k}.$$
zhw.
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