Riemann's rearrangement theorem

File: Rearrangement of series. The theorem of Levy-Steiniz. - Google Chrome.jpg (23KB, 1108x201px) Image search: [Google]

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So for our partial sum Sn, we can get any real number depending on the order we sum the terms of the series as long as the sum of positive and the negative terms separately diverge.
But there seems to be a fundamental point missing here. I get that for example for the series in the pic we can reach past 10 just by adding a number of positive terms, then add one negative and continue to oscillate around 10.
Yes, but we are essentially assigning a limit to the partial sum of the series to determine the convergence. However limit assumes an inherent "after somehow completing the impossible task of reaching infinity".
Take the infinite sum of the series (1/2)^n for example. We could write the partial sum formula for this as:
1/1/2^n, and its limit as n tends to infinity is 1. We can see that no matter the n, the actual value will never be 1 though, so the limit operation is a convention assuming the complition of the infinite steps.

I mean, exactly as many unused/extra negative terms are stacking up in the alternate harmonic series as we use up the positive terms and we, somehow completing this infinite sum implies that we have to use all the terms we had, which would return us to an S of absolute convergence, not a conditional Sn.