Amusingly enough, in the 2-adics (base 2 p-adic integers) the series 1+2+4+8+... converges to the number -1.
The sum is "obviously" the 2-adic element ...1111. Which, as neil correctly points out, is exactly -1 in 2's complement notation. Or just add 1 and see you get .
And it is not a problem, as 2-adics aren't ordered.
Of course, if you want to work with real number
s, you have to give up on this p-adic
nonsense. And then you're back to square 1
: 1+2+4+... doesn't converge to any real number!