1 + 2 + 3 + 4 + 5 + 6 +.... = -1/12
(via: Wimp.com) This simple version of the proof does have a bit a of a fudge, because the sum
only equals 1/(1-x) for the absolute value of |x| < 1, not (as the video stated, for x < 1). There's a longer version of this proof on the Numberphile Web site, using the Riemann Zeta function and what's called analytic continuation:
The thing is, this analytic continuation makes both mathematical and physical sense -- physics experiments have confirmed it. For example, the well-known Casimir effect, which is the electrical attraction of two parallel, uncharged metal plates due to quantum effects, is calculated to involve the Zeta function ζ(-3), which by similar reasoning equals 1/120, and the predicted Casimir force has been verified by experiments. That is
is experimentally verified.
Wikipedia has an entry on this sum, including a brief one on its relationship to string theory, and this wonderful excerpt from a letter from the Indian genius Srinivasa Ramanujan's to the mathematician who eventually brought him to Englaand, G. H. Hardy:
"Dear Sir, I am very much gratified on perusing your letter of the 8th February 1913. I was expecting a reply from you similar to the one which a Mathematics Professor at London wrote asking me to study carefully Bromwich's Infinite Series and not fall into the pitfalls of divergent series. … I told him that the sum of an infinite number of terms of the series: 1 + 2 + 3 + 4 + · · · = −1/12 under my theory. If I tell you this you will at once point out to me the lunatic asylum as my goal. I dilate on this simply to convince you that you will not be able to follow my methods of proof if I indicate the lines on which I proceed in a single letter. …"