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You keep saying this sum is 'wrong.' Of course, it's not. Obviously, if you EVER stop at any point while taking this sum, you'll end up at a positive, finite, and possibly massive number. But the point is, your are taking the sum to infinity, meaning that there ISN'T a stopping point. The sum of all positive integers is 1/12, and it's been proven (the zeta function is real and completely valid); it's not 'wrong' just because it doesn't make sense.
Also, (interesting but perhaps unrelated) if you take the ideas used to calculate this sum, you can find that the sum of all odd positive integers is +1/12 the sum of all positive even integers is 1/6.