convergent of series problem So I have this question which I do not understand:
The series $ \sum_{n=1}^{\infty} |a_n| $ is convergent.
The series $ \sum_{n=1}^{\infty} b_n $ is convergent, but the series $ \sum_{n=1}^{\infty} |b_n| $ is not convergent.
So which one of the following is not always true:


*

*You can change the order of the sequence so that $ \sum_{n=1}^{\infty} (a_n + b_n) $ so it will convergent to $0$

*You can change the order of the sequence so that $ \sum_{n=1}^{\infty} (a_n b_n) $ so it will convergent to $0$

*Each brackets you will put on  $ \sum_{n=1}^{\infty} (a_n b_n) $ will be convergent

*Each brackets you will put on  $ \sum_{n=1}^{\infty} (a_n + b_n) $ will be convergent


Could someone give me a counterexample for this question ?
I tried to solve that but I did not make it.

We know
$$\sum_{n=1}^\infty a_n\;,\;\;\sum_{n=1}^\infty |a_n|\;,\;\;\sum_{n=1}^\infty b_n\;\;\text{are convergent}\;,\;\;\sum_{n=1}^\infty |b_n|\;\;\text{is divergent}$$
From here and from arithmetic of limits (of series), we get at once that (1) is  always true since this is a conditionally convergente series (why?) and we apply Riemann's Theorem...and this already gives you the answer: (4) is not always true.
