The sum of an infinite series of positive numbers is negative

let s={1+2+4+8+...}
then s-1={2+4+8+16+...}
and 2s={2+4+8+16+...}
by inspection, s-1=2s
which gives us s=-1

in conclusion, {1+2+4+8+...}=-1












the mistake is that{1+2+4+8+...}=infinity

and infinity-1=infinity, and infinity*2=infinity

but not all infinities are equal

This is simply proof that the world of mathematics uses two's complement notation :)

small's analysis of the mistake is correct, except I don't like eir use of the term `infinity'. It is probably better to say that the sum of the series { 1 + 2 + 4 + 8 + . . . } does not exist. Otherwise one can get into nonsensical arguments about ``what is infinity minus infinity?''.

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.

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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!

This sentence could be a reference to the genius mathematician Srinivasa Ramanujan. He famously taught himself mathematics in India. With little access to texts, his notation had evolved to suit his own thoughts and so was completely incomprehensible to the Western mathematicians to whom he sent his work. The series he sent summed to -1/12, and was noticed by mathematicians G.H. Hardy and Littlewood to be an important result in the context of the Riemann Zeta Landscape.

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The result itself was as follows:

Zeta(-1) = 1 + 1/(2^-1) + 1/(3^-1) + 1/(4^-1)... Which Ramanajun calculated to be equal to -1/12. The dismissal he recieved from most western mathematicians was due to the fact that, as the strangeness of this node-title suggests, he wrote this out as:

1 + 2 + 3 + 4 + ... = -1/12

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Further messages or additional writeups in this node would be welcome. I'm not a professor of mathematics, which I would need to be to do this topic justice.