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If $b_n$ is decreasing and positive and $\sum b_n$ converges and $1/a_n$ is decreasing and positive and $\sum 1/a_n$ diverges, must $\lim \limits_{n \to \infty} a_n b_n=0$?

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merged. For future reference, being rude makes me no more likely to help you out. –  Qiaochu Yuan Dec 10 '11 at 19:59

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No. Fix $b_n$, WLOG we can assume $B_n = b_n^{-1}$ are integers.

Define $a_n$ as follows inductively. Let $k_1 = 1$ and $k_{j+1} = B_{k_j}^2 + k_j$. And between $k_j$ and $k_{j+1}$, define $a_k = B_{k_j}^2$. Then clearly $a_n^{-1}$ is a divergent series.

But $\limsup a_n b_n \leq \limsup B_{k_j}^2 b_{k_j} \nearrow \infty$.

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