Confusion in conditional convergence

Question

As we know if a series is conditionally converges to a some finite value L , we can change this value(L) of convergence to some other other value just by changing the order of series .

But, is it possible to change the order in such a way that the new series becomes divergence?

and if it is not possible then how to prove it ?

Answer 1

Hint

Yes this is possible. The idea is that as $\sum \vert a_n \vert$ diverges, you can add enough positive terms in order for $\sum a_{\sigma(n)} $ to be larger than any $N \in \mathbb N$, bring only a negative term and so on.

Answer 2

Here: https://en.wikipedia.org/wiki/Riemann_series_theorem

you can find a proof for the existence of a rearrangement that diverges to $ \infty.$