The sum of an infinite series of positive numbers is negative (idea) by ariels

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 0.

And it is not a problem, as 2-adics aren't ordered.

Of course, if you want to work with real numbers, 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!

False mathematical proofs	2^.5 = 2	p-adic integer	386 overclocking for newbies
Infinity bottles of beer on the wall in c++	e	Infinity is where parallel lines cross	Converges to the Tao at infinity
Guido Ubaldus' proof of the existence of God	the sum of all positive even numbers is -2	Memorizing Pi	twos complement
infinity bottles of beer on the wall	Ramanajun	Since time T equals minus infinity	p-adic
Proof by oddity	credit card validation	Srinivasa	Two's Complement
1/infinity	Infinity plus one	Soylent Green	Riemann zeta function