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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.2012-06-27
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    @MarkDominus Yes. Just to start my $n$ from $0$.2012-06-27
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    Hmm, the two series are only conditionally convergent. Are there also examples where they are absolutely convergent?2014-07-16
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    @GottfriedHelms: [see this](http://math.stackexchange.com/questions/133400/what-are-the-rules-for-convergence-for-2-series-that-are-added-subtracted-multip)2016-11-21