Why the sum of all positive integers is -1/12


To quote Talking Barbie: “Math is hard! Let’s go shopping!”


No, this is not a proof. Sorry. By violating the rules you can “prove” all kinds of stuff, like that 1 is equal to zero (usually by hiding the fact that you divided by zero), and with many divergent series you can produce any sum you want by fooling around with the order of the terms.


Or not. They are using a description of a series and pretending it is actually the sum as MarkCC points out here: http://scientopia.org/blogs/goodmath/2014/01/17/bad-math-from-the-bad-astronomer/


What bullshit.


I was going to complain how adults prefer to get information like this written down, rather than in a video format, but seeing as it’s all apparently bullshit anyway… thanks for sparing me that?


As soon as they said the sum of an undefined series was not undefined, I stopped watching.

But I shouldn’t be at all surprised if they end up deriving a counterintuitive result from a bad premise.


Basically, this sum is undefined so it could have several answers. However, in many physics applications they know that this sum has to come out to a finite number so they are able to give a finite part of -1/12. A more intelligent discussion is here: http://math.stackexchange.com/questions/39802/why-does-123-dots-1-over-12


Isn’t this an example of reductio ad absurdum?


much better more detailed video here https://www.youtube.com/watch?v=E-d9mgo8FGk

One thing I’m thankful for these days: two or three years ago, there would have been a retraction followed by Comments are closed.


Sounds like integer overflow to me… finally, proof the universe is a computer.

Or it could just be bullshit.


Shane, I think both those things would be an improvement: a retraction explaining why the original submission was bullshit and an end to comment-pile-on. The actual problem with the old days wasn’t that there’d be a retraction and no comments, the article would simply be deleted with no acknowledgment of the dumbness. What you’ve described should be the future. In the absence of a correction and a lack of comments, transparency and comment-pile-on is an acceptable substitute.


Yeah, any time I see a negative result of adding positive numbers I can’t help thinking “you spilled into the sign bit! Should have used a bigger int var!” automatically. But not this time…


Well, everything he did is false, yes, because he’s talking about series that diverge. But also he’s not talking about the limit of the series, he’s talking about the analytically continued zeta function, evaluated at s=-1, giving -1/12 as noted above. He just didn’t say so, which is basically irredeemable.

I think that this video is utterly sad, because the takeaway message is “wow, that’s cra-zy! MIND BLOWN”, but still nobody understands anything. Imagine taking a tour of an art gallery, and the tour guide is going “Guys? GUYS. HOLY. CRAP. LOOK AT ALL THIS FECKIN PAINT! And someone just nailed it to the WALL! MIND BLOWN!”


Since I’m a mathematician I feel obligated to say two things:

  1. The sum of all positive integers isn’t -1/12. That sum does not converge.
  2. There is a way of making sense of this formal sum which gives it the value of -1/12. That value isn’t unique and calling this value the “sum of the positive integers” is at best weasel wording.

Here is a better explanation: In calculus, you learn that the harmonic series:
\sum_{n=1}^{\infty} 1/n
diverges. Let’s change this a bit to get a function R(s):
R(s) = \sum_{n=1}^{\infty} 1/(n^s)
This is the famous “Riemann zeta function” and the harmonic series is R(1).

Now what about R(-1)? Well, if you just look at the definition you would see that
= \sum_{n=1}^{\infty} 1/(n^-1)
= \sum_{n=1}^{\infty} n
i.e. the sum of all positive integers. The thing is, if you tried to graph this definition of R(s) around s=-1, you wouldn’t get anything at all. However, if you started in a place where the definition did make sense (s>1) and tried to extrapolate the values the values from those (analytic continuation), then you would see a continuous function with value -1/12 at s=-1.

This depends on identifying the sum of all positive integers with a value of the Riemann zeta function. You could easily pick something else and get a different value.


Sorry, but you can only say that “1 + 2 + 3 + 4 + … = -1/12” by changing the definitions of “+” and “=”, I’ve tried to explain why the equation as stated is false here.


I see what you’re saying, and you’re right: there are definitely situations where stories were just disappeared, sometimes large volumes of stories without comment cough. On the other hand, although there’s a pile-on here, there’s also a discussion going on. I figured it had to be B.S., and thankfully there are people here qualified to explain it to people like me. :->

My favorite fanciful math video is the Wau Number. How quickly will you figure it out?


Worse, with some convergent** series you can get any number you want by rearranging the terms.

** They have to be conditionally convergent. i.e. when you take absolute values of the terms, the sum doesn’t converge.

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