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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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Isn't a more appropriate place to seek an answer for this question ? – Clyde Lobo Mar 9 '11 at 7:44
What is this website specifically for? – Daniel Mar 9 '11 at 7:47
@Daniel: Well... In that case why don't you just email him instead of posting it over here? – user17762 Mar 9 '11 at 8:17
Not that tough... But fun! :) – Hans Lundmark Mar 9 '11 at 8:34
The thing about questions like these is that your brain always assumes that sequences are more or less monotonic, so you have to actively work against this tendency. – Qiaochu Yuan Mar 9 '11 at 11:54
up vote 8 down vote accepted

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.

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Great! Thank you! – Daniel Mar 9 '11 at 8:37

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