Recently, a video posted on Numberphile titled, Astounding Result, has been making rounds on the web. Of course, this video has resulted in a plethora of criticisms, and I would argue some of it is justified. I wasn’t convinced either and had my skepticisms as well. However, I continued to research the problem and tried to gain a greater understanding. What I concluded is that the result isn’t necessarily wrong, but they did it improperly. The way they present the problem and the terminology they use are both improper.
EDIT: Numberphile posted an excellent follow-up video on March 18 titled, Why -1/12 is a gold nugget. For those interested, I highly recommend watching it.
Early on in the video, they display this result in a “well-known string theory book,” presumably to give it some justification. If you look closely, this is the way in which the result is presented in the book:
Notice that this is not an equation as there’s no equality = sign. Rather, what you have is an arrow, which means something quite different. That’s the first mistake they made in the video, using the equality sign everywhere as to signify that this summation “equals” -1/12, which is not how it’s understood in mathematics or physics.
Secondly, if you pause the video on a stable frame (where the camera isn’t jittering), you can clearly make out the preceding text in the book. To save you the trouble, this is what is written:
It can be evaluated by regulating the theory and then being careful to preserve Lorentz invariance in the renormalization.
There are a few special terms here which are, unfortunately, never used in the video. This is the second mistake they made in the video, the lack of proper terminology. For those of us who studied Calculus in college, our natural instinct is to test if the sequence converges on some finite value L. If not, then we simply conclude the series diverges and move on with our day. However, they only taught us one method of summation, otherwise known as Cesàro summation. When using this summation method, the series does indeed diverge. However, there are other techniques for “assigning a meaningful value to a divergent series,” collectively known as regularization (as well as renormalization, used in quantum mechanics).
There was further criticism in how they proved these results, by taking these infinite sequences and adding them together, multiplying, shifting to the right, etc. I’d rather not get into this much. Instead, I’ll just point you to the Riemann series theorem…
In the video and other places on the web, I’ve seen the claim that these results are used in “many areas of physics” and “have been observed in nature.” The most common example I’ve encountered relates to string theory, which I think is moot because, as I understand it, string theory hasn’t been tested experimentally (i.e. it’s untestable using existing equipment) and is therefore little more than just theory. However, in an alternate video proof , a reference is made to the Casimir force (i.e. the Casimir effect), which has been measured experimentally.
I’m not claiming to have any deep understanding of the mathematics or physics presented here. I simply felt that clarification is needed in order to settle the uproarious masses. I may investigate this topic further in the future, but for now, I’ve satisfied my skepticism. In conclusion, the result isn’t necessarily incorrect (I’ll leave it to the masses to disprove), but greater care could have been taken in the presentation, the proofs, and the terminology used.
There’s a second, much longer and more complex proof presented in this video,  Sum of Natural Numbers. The Wikipedia article covering this sequence can be found here (1+2+3+4+5+6…). Another interesting sequence covered in much more detail can be found here (1-2+3-4+5-6….). The Wikipedia article for divergent series contains some useful information as well, including some of the methods used for summation. A couple of the highly critical responses prompted by this video: One and Two.