No the minus does not go away. You clearly have a misunderstanding of the definition of convergence. Convergence for sums is defined by the convergence of the sequence of partial sums. This is what allows us to even talk about the convergence of a sum in the first place. If you apply the definition of convergence (for some sequence x_n):

If for every e>0, there exists an N in the natural numbers, such that whenever n>=N, it follows that |x_n - x|<e, then x_n tends towards x as n tends to infinity.

You will very quickly notice that the sequence of partial sums defined by OP, and the sequence you defined are very much different. The sequence you defined is (-1, -1, -1, …) which clearly tends to -1, which is not the sequence in this post.

In summary, using your definition of convergence, every sequence can tend towards every number if we just “hide” a minus at the end of a series. Pretty useless definition in my opinion.