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Can anyone think of sequences $\{a_n\}$, $\{b_n\}$ such that $\sum a_n$ diverges, ${b_n}\to\infty$, but $\sum a_nb_n $ converges?

Thank you.

Note that $\{a_n\}$ must have infinitely many positive terms and infinitely many negative terms.

Edit: I get the feeling that only Qiaochu Yuan could answer this one... ;)

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    I knew that $b_n$ could not be monotonic, and I knew that $a_n$ had to oscillate in a way such that $a_nb_n$ would oscillate in a convergent matter, but could not think of an answer :( very happy to have an answer, though.2011-03-09

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Let $c_n = (-1)^n/\sqrt{n}$. Then $\sum c_n$ converges by the alternating series test. Let $b_n=\sqrt{n}$ or $n$ depending on whether $n$ is even or odd. Then $b_n \to \infty$. Let $a_n = c_n / b_n$. Then $a_n = 1/n$ if $n$ is even and $a_n = -1/n\sqrt{n}$ if $n$ is odd, so the negative $a_n$'s have a finite sum, which the positive $a_n$'s don't. Hence $\sum a_n$ diverges.