Specify whether the series $sum_{n=n_0}^{+ infty } a_n cdot b_n$ must be convergent in cases a) or b)
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Specify whether the series $sum_{n=n_0}^{+ infty } a_n cdot b_n$ must be convergent when: a) $sum a_n$ convergent, $sum b_n$ convergent b) $sum a_n$ convergent absolutely, $sum b_n$ convergent. I need to check my reasoning and I need help to guide them further: If $sum b_n$ convergent, then in particular a sequence of partial sum $sum b_n$ is convergent, so the second condition of Dirichlet's test is met. That is why we should take care of $a_n$ . The first condition of Dirichlet's test is $a_n$ monotonic and $a_n$ convergent to $0$ . So I think in the case of a) the series $sum_{n=n_0}^{+ infty } a_n cdot b_n$ not always is convergent and b) must be convergement, but there are only my thoughts and I do not know how to prove it.
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