First note that you can apply this technique only to so-called conditionally convergent series, i.e. series that are not absolutely convergent. If a series if absolutely convergent, then any permutation on the order of summands still converges, and to the same limit.

If you're startled by the length of any rigorous proof to Riemann's rearrangement theorem, there's no need to be - it all boils down to this key concept: One can prove that if a series is conditionally convergent, then the sum of all its positive terms tends to infinity, and the sum of all its negative terms tends to infinity. Suppose you have two infinite bags of such terms (the positive and the negative terms respectively of some conditionally convergent series), and you wish to find a permutation that makes your series converge to some desired limit . Well, start adding up positive terms until you've surpassed by a little bit; then stop, and start adding negative terms until you're below by a little bit; and so forth. When taking the limit of this process it's not hard (although a bit technical) to verify that the sequence of partial sums stabilizes around , converging to it as a limit.

It's also possible to make your series diverge to pos/neg infinity this way, and also to make it diverge altogether (by letting it oscillate between 1 and -1 for example).