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I am studying the behaviour of product of a convergent and a divergent infinite series. I found a example in which product series come out to be a convergent series . But can't get a divergent series Does product series always comes out to be convergent ? Or it can be divergent, ? Please give your views on it

Shubham singla
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  • What's your example? – Matthew Leingang May 13 '20 at 18:48
  • I take series 1/n which diverges... And series 1/n^2 which converges and product converges – Shubham singla May 13 '20 at 18:51
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    It certainly is not the case that the product series always converges. Take $a_n = n$ and $b_n = \frac 1 {n^2}$ Then $\sum a_n$ diverges, $\sum b_n$ converges and $\sum a_nb_n$ diverges. You can make weaker statements though. For example, if $\sum a_n$ converges absolutely and $b_n \to 0$ then $\sum a_n b_n$ converges. – User8128 May 13 '20 at 18:51

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No it isn't always convergent. (I assume by "product you mean $\sum a_nb_n$.)

Take $\sum\frac{(-1)^n}n$ as one of the series and $\sum(-1)^n$ as the other.

saulspatz
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