Since the issue of the OP seems to have been settled amicably, I thought I would continue this interesting sidelight. It turns out that 2-adic arithmetic (and p-adic arithmetic for other primes p) is interesting to topologists as well as number theorists. And Orbifold is entirely correct that it is the limit of modular arithmetic (in modern terminology; in slightly older terminology it is the inverse limit). One way to look at it is to consider infinite sequences (a[sub]1[/sub], a[sub]2[/sub], a[sub]3[/sub], …) where each a[sub]n[/sub] is an integer modulo 2[sup]n[/sup] and a[sub]n-1[/sub] is congruent to a[sub]n[/sub] modulo 2[sup]n-1[/sup]. (That is, it is the remainder on dividing a[sub]n[/sub] by 2[sup]n-1[/sup].) If you were to stop your sequences at the nth term, looking at only (a[sub]1[/sub], a[sub]2[/sub], …, a[sub]n[/sub]), then the whole sequence would be determined by the last term, a[sub]n[/sub], and you would be looking at just arithmetic modulo 2[sup]n[/sup]. However, when you “take the limit” and allow yourself to look at infinite sequences, you get a new beast, the 2-adic numbers.
Now write 1 for (1, 1, 1, …), 2 for (0, 2, 2, 2, …), 4 for (0, 0, 4, 4, 4, …) and so on. Then 1 + 2 + 4 + 8 + … = (1, 3, 7, 15, …) = (-1, -1, -1, …) = -1. 
And let me repeat Orbifold’s warning: This has nothing to do with infinite sums of real numbers. In fact, it’s almost the opposite, in some sense. For those mathematicians in the audience worried about the infinite sums above, there is a topology around (even a metric, if I remember correctly) in which those sums converge. In fact (again, IIRC) we’re looking at the completion of the integers with respect to this metric. But that point of view obscures the connection with modular arithmetic.