I understand that given the absolute convergence test, if I am able to prove that the absolute of the series converges, then the series itself will converge itself as well.
What if I want to prove for conditional convergence? Is it sufficient to prove straight away that the absolute does not converge, or do I have to first prove that the series converges, and then prove that absolute does not converge, hence it must converge conditionally?
On
A series converges conditionally if both of the following conditions are true:
To prove a series converges absolutely, you need to prove both conditions. It is not enough to prove that the series does not converge absolutely, since, for example, $$\sum_{n=1}^\infty (-1)^n n$$ does not converge absolutely, but that doesn't mean it converges conditionally.
Yes, of course you need to prove both. If you prove the series of absolute values does not converge it doesn't mean that the original series converge. For example the series $\sum_{n=0}^\infty (-1)^n$ simply diverges.