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

Here's a brain-meltingly cool proof of the bizarre mathematical truth that the sum of all positive integers (1 + 2 + 3 + 4 + 5....) is -1/12. This is not only provably true, it's also foundational to certain testable elements of physics. In other words: not just a logical curiosity, but also the bedrock of real-world, useful stuff.

## Notable Replies

1. To quote Talking Barbie: "Math is hard! Let's go shopping!"

2. joe_b says:

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.

3. 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?

4. 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.

5. knappa says:

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
R(-1)
= \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.

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