Numberphile v. Math: the truth about 1+2+3+...=-1/12

Yes, I kept waiting for them to "fess up" at the end but they didn't. Mathologer called them out on it. ▶ this video

Just spent 5 minutes making a script to find all the links in this thread and then 'score' them on the number of votes. Here's all the channels with 2 or more: 6 Votes: link link 5 Votes: link link link link 4 Votes: link 3 Votes: link link link link link link link link link link link link link link 2 Votes: link link link link link link link link link link link link link ▶ this video link link link link link link link link link link link link link link link link link link link link link link link link link link link link link link link link link link link link

It's similar. Two issues with Numberphile videos I can remember: Their video on 1+2+3+... = -1/12 was so bad that another Math channel (Mathologer, which is highly recommendable btw) did a whole 40min video tearing it apart: ▶ this video In their video on fluid mechanics, they say that the Navier-Stokes equations describe all fluids on earth, while they only show the incompressible version (without clarifying). The examples shown on screen at that moment also include a Ketchup bottle, and they don't discuss non-Newtonian fluids either (Ketchup is non-Newtonian, that's why you can get a lot of it coming out at once if you shake a bottle too hard). It's a bit nitpicking though, because apart from these missing caveats it's a great video (with a guest lecturer from Tom Rocks Maths, another great math channel).

I know you're asking about Computerphile, but this video explains how one of the Numberphile videos is wrong: ▶ this video

I find a response video to it more enlightening. It's often found as the number one related video. ▶ this video

That proof (and the more general statement) is not correct because it makes assumptions about infinite summation of divergent series that are not true. There is a link between the sum of natural numbers -- namely, the analytic continuation of the Reimann Zeta function has Z(-1) = -1/12. But the Zeta function is only defined to equal the sum of inverse powers for Re(z) > 1. And under a specific definition of summation (Ramanujan summation) you can say that "the sum is equal to -1/12" but that isn't the same as normal summation. Mathologer did a fairly in-depth video[1] into why this proof is wrong and what the actual link is between the infinite series and -1/12. The upshot is that even if you use more complicated definitions of summation, you cannot define sums of the kind 1+2+3+... to equal a finite number. If you assume the infinite sum of 1+2+3+... converges to a finite number you can easily prove a contradictory statement using the same summation properties assumed by that proof. Namely: S = 1 + 2 + 3 + ... = -1/12 S2 = S-S = 0 = 1 + 2 + 3 + ... - 1 - 2 - 3 - ... = 0 = 1 + 2 + 3 + ... - 1 - 2 - ... = 0 S2 = 1 + 1 + 1 + 1 + ... = 0 S3 = S2-S2 = 0 = (1-1) + (1-1) + ... = 1 - 1 + 1 - 1 + ... = 0 But we "derived" in the original proof that S3 is equal to 1/2, which is a contradiction. (You could derive S3 is 1/2 from S but that's what the article does in reverse.) [1]: ▶ this video

Burkard Polster's (Mathologer) videos on the subject are probably going to be more useful than Numberphile. Numberphile merely presents a trick; Mathologer points out both that it's patent nonsense [0] and that it's useful patent nonsense [1]. [0] ▶ this video [1] link