Its a conditionally convergent series; you can re-arrange the terms to add up to any real number you like (or diverge). [1].

[1]http://en.wikipedia.org/wiki/Conditional_convergence

You can construct a new series such that every term in the old one is there. However, if you try and rearrange the terms you end up with a backlog that coverages to the difference between the series. Which is why the actual series converges in the first place.