Does it make mathematical sense to do an absolute convergence test if the original series diverges? Reason I ask I know a series can converge but then when you apply the absolute convergence test it may diverge.  I understand this part.  One concludes absolute convergence is a stronger condition! 

But what happens if the original series diverges and the terms are negative , how do I know that by making it positive it won't become convergent?  In this scenario you would never know that absolute convergence was the strongest condition.   

The solution would be that it makes no mathematical sense to apply absolute convergence to a divergent series.  Or is this just by definition maybe? 

Note that if $\sum a_n$ diverges, with $a_n$ negative, also $\sum |a_n|=-\sum a_n$ diverges.
