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?
Example of two convergent series whose product is not convergent.
9
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calculus
sequences-and-series
1 Answers
17
$$a_n = b_n = \dfrac{(-1)^n}{\sqrt{n+1}}$$ where $n \in \{0,1,2,\ldots\}$
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1Why 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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1@MarkDominus Yes. Just to start my $n$ from $0$. – 2012-06-27
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1Hmm, the two series are only conditionally convergent. Are there also examples where they are absolutely convergent? – 2014-07-16
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1@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