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Could someone give me an example of two convergent series $\sum_{n=0}^\infty a_n$ and $\sum_{n=0}^\infty b_n$ such that $\sum_{n=0}^\infty a_nb_n$ diverges?

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$$a_n = b_n = \dfrac{(-1)^n}{\sqrt{n+1}}$$ where $n \in \{0,1,2,\ldots\}$

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Why did you put $\sqrt{n+1}$ in the denominator instead of $\sqrt n$? It seems to me that it would work either way. – MJD Jun 27 '12 at 18:23
@MarkDominus Yes. Just to start my $n$ from $0$. – user17762 Jun 27 '12 at 18:51
Hmm, the two series are only conditionally convergent. Are there also examples where they are absolutely convergent? – Gottfried Helms Jul 16 '14 at 7:36

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