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Give $\sum a_n$ diverges and $\sum b_n$ converges conditionally, if the Cauchy product $\sum c_n$ must be divergent?

Xin Fu
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  • Is $a_n$ and $b_n$ positive ? – Chinnapparaj R Jun 29 '19 at 02:03
  • @ Chinnapparaj R , not necessary. If $a_n$ and $b_n$ are positive, then $\sum c_n=\sum_{n=0}^{\infty}\sum_{k=0}^na_kb_{n-k}\geq \sum_na_nb_0$, so it diverges. – Xin Fu Jun 29 '19 at 02:30
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    @ChinnapparajR: Since $\sum b_n$ converges conditionally, we can't have $b_n\geq0$ for all $n$, otherwise it would converge absolutely. – Clayton Jun 29 '19 at 02:44

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It is possible for a divergent series $\sum a_n$ and a conditionally convergent series $\sum b_n$ to have a Cauchy product $\sum c_n$ that is not only not divergent, but actually absolutely convergent. A rather complicated example is constructed in this paper.

Surprisingly, it is somewhat less difficult to find two divergent series whose Cauchy product is convergent, as I show here.

RRL
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