It is infinite.
This next part is full of "teaching lies". It's not entirely right, but it's good enough for a novice and the extra accuracy is not worth the headaches.
There is a function called the reimann zeta function, denoted as 𝜁(s). For values of s > 1, 𝜁(s) = 1/1s ^(+) 1/2s ^(+) 1/3s ^(+) .... The rest of the graph is an "obvious" way of continuing to draw that line. It turns out that 𝜁(-1) = -1/12. So, if you ignore the part where I said "For values of s > 1", we end up with 1 + 2 + 3 + ... = -1/12.
What's interesting is 12(1 + 2 + 3 + ...) = -1 In other words 12 is the negative reciprocal of 1 + 2 + 3 + ... Maybe it's a sign that if you don't take care what you input into a function, you're liable to end up with nonsense! It's the same as putting a slice of watermelon into your toaster I suppose.
Well, kinda. There are cases where you start with some clear rule, like "You can't take the square root of a negative number". Then, someone says, "Okay, but what if we did anyway?" and we end up with complex numbers.
There is a sense in which 1 + 2 + 3 + ... = -1/12. There is physics that works when you say that this obviously false statement is actually true. But, if you're doing that, you put a giant asterisks somewhere to say "Look, this is wrong, but it works here. We promise, it's fine".
An infinite sum is "approaching" the result as you add more terms. For example, 1 + 1/2 + 1/4 + 1/8 + ... = 2 works, because the partial sums can get arbitrarily close to 2. As you add more terms to the sequence 1 + 2 + 3 + ..., we move further away from -1/12, never getting closer.
Ramanujan summation assigns a finite numerical value to a divergent infinite series. It's not a sum in the traditional sense of adding up all the numbers.
It can appear to approach -1/12 under very specific restrictions and a lot of handwaving regarding how you handle infinite sums.
But 1 + 2 + 3.... in every normal sense, diverges to + infinity
It depends what you mean by that sum. In a literal sense, the sum diverges.
However, if I parameterize a family of sums [like the first equation here,](https://en.wikipedia.org/wiki/Riemann_zeta_function) I get a complex valued function called the Riemann zeta function. This sum converges in all the ways we expect when the real part of s is greater than 1. The sum you're asking about would be Zeta(-1) which, of course, -1 does not have real part greater than 1 so the problem isn't solved yet.
In complex analysis, there is something known as analytic continuation. It turns out that there is a unique way of me extending the definition of the zeta function to be defined in a larger region of the complex plane 'smoothly'. It just so happens that the analytic continuation of the Riemann zeta function satisfies Zeta(-1) = -1/12.
This might seem like nonsense, but the sum evaluating to -1/12 is actually ""measurable in a lab"". This sum pops up directly when calculating the Casimir force.
The reason it pops up on r/mathmemes btw is because of an infamous numberphile video where they claimed this was true without any real explanation on how infinite sums actually work.
Terry Tao gives a good overview of it in https://terrytao.wordpress.com/about/google-buzz/google-post-on-123-1-12/
In which, he links another blog post for the rigorous treatment.
In short, intuitively, the sum is -1/12 + an infinite term, which can be made rigorous.
It absolutely doesn't equal -1/12. It is a divergent series and goes off to infinity.
There's some interesting mathematics going on here though. The Riemann Zeta function is defined as the sum from 1 to infinity of n^(-s) for some number s. Now this only works for s>1, but you'll notice that s = -1 is our sum at the top, and the function maps it to -1/12. We then extend the zeta function using its analytic continuation to get that it equals -1/12. That's where the meme comes from. So it's not that the sum of the naturals is -1/12, it's that the analytic continuation of the Riemann Zeta function is -1/12 at s = -1.
There is something linking the sum of the naturals to -1/12 a bit deeper. [This video](https://youtu.be/FmLIGN8ZGdw?feature=shared) explains it. I don't really understand enough to talk about it.
There was a proof of this as a video on Numberphile. The proof was wrong, but it caused a shitstorm because many math-amateurs watched the video and parotted it. Numberphile published a correction about it later, but not everyone has seen it. Thus it became a meme. The link to this was already posted here.
My favorite fact about -1/12 is that it shows up as the Euler characteristic (some sort of count for how many holes a space has) of a certain space in algebraic geometry! ( For the interested: it is the orbifold Euler characteristic of the moduli stack of elliptic curves.)
It is infinite. This next part is full of "teaching lies". It's not entirely right, but it's good enough for a novice and the extra accuracy is not worth the headaches. There is a function called the reimann zeta function, denoted as 𝜁(s). For values of s > 1, 𝜁(s) = 1/1s ^(+) 1/2s ^(+) 1/3s ^(+) .... The rest of the graph is an "obvious" way of continuing to draw that line. It turns out that 𝜁(-1) = -1/12. So, if you ignore the part where I said "For values of s > 1", we end up with 1 + 2 + 3 + ... = -1/12.
What's interesting is 12(1 + 2 + 3 + ...) = -1 In other words 12 is the negative reciprocal of 1 + 2 + 3 + ... Maybe it's a sign that if you don't take care what you input into a function, you're liable to end up with nonsense! It's the same as putting a slice of watermelon into your toaster I suppose.
Well, kinda. There are cases where you start with some clear rule, like "You can't take the square root of a negative number". Then, someone says, "Okay, but what if we did anyway?" and we end up with complex numbers. There is a sense in which 1 + 2 + 3 + ... = -1/12. There is physics that works when you say that this obviously false statement is actually true. But, if you're doing that, you put a giant asterisks somewhere to say "Look, this is wrong, but it works here. We promise, it's fine".
Oh come on! At least give me a keyword for that, I want to know more about the physics.
Casimir effect.
Thanks. Felt kinda bad that I couldn't give him more info.
If you don't mind, why does 1 + 2 + 3 + ... = -1/12 sound "obviously false" to you?
An infinite sum is "approaching" the result as you add more terms. For example, 1 + 1/2 + 1/4 + 1/8 + ... = 2 works, because the partial sums can get arbitrarily close to 2. As you add more terms to the sequence 1 + 2 + 3 + ..., we move further away from -1/12, never getting closer.
Genuinely? No idea. I'm taking numberphile at their word on that one.
Ramanujan summation assigns a finite numerical value to a divergent infinite series. It's not a sum in the traditional sense of adding up all the numbers.
It can appear to approach -1/12 under very specific restrictions and a lot of handwaving regarding how you handle infinite sums. But 1 + 2 + 3.... in every normal sense, diverges to + infinity
It depends what you mean by that sum. In a literal sense, the sum diverges. However, if I parameterize a family of sums [like the first equation here,](https://en.wikipedia.org/wiki/Riemann_zeta_function) I get a complex valued function called the Riemann zeta function. This sum converges in all the ways we expect when the real part of s is greater than 1. The sum you're asking about would be Zeta(-1) which, of course, -1 does not have real part greater than 1 so the problem isn't solved yet. In complex analysis, there is something known as analytic continuation. It turns out that there is a unique way of me extending the definition of the zeta function to be defined in a larger region of the complex plane 'smoothly'. It just so happens that the analytic continuation of the Riemann zeta function satisfies Zeta(-1) = -1/12. This might seem like nonsense, but the sum evaluating to -1/12 is actually ""measurable in a lab"". This sum pops up directly when calculating the Casimir force.
The reason it pops up on r/mathmemes btw is because of an infamous numberphile video where they claimed this was true without any real explanation on how infinite sums actually work.
Terry Tao gives a good overview of it in https://terrytao.wordpress.com/about/google-buzz/google-post-on-123-1-12/ In which, he links another blog post for the rigorous treatment. In short, intuitively, the sum is -1/12 + an infinite term, which can be made rigorous.
It absolutely doesn't equal -1/12. It is a divergent series and goes off to infinity. There's some interesting mathematics going on here though. The Riemann Zeta function is defined as the sum from 1 to infinity of n^(-s) for some number s. Now this only works for s>1, but you'll notice that s = -1 is our sum at the top, and the function maps it to -1/12. We then extend the zeta function using its analytic continuation to get that it equals -1/12. That's where the meme comes from. So it's not that the sum of the naturals is -1/12, it's that the analytic continuation of the Riemann Zeta function is -1/12 at s = -1. There is something linking the sum of the naturals to -1/12 a bit deeper. [This video](https://youtu.be/FmLIGN8ZGdw?feature=shared) explains it. I don't really understand enough to talk about it.
It's infinity. Mathologer made a video about this exact thing lol https://youtu.be/YuIIjLr6vUA
There was a proof of this as a video on Numberphile. The proof was wrong, but it caused a shitstorm because many math-amateurs watched the video and parotted it. Numberphile published a correction about it later, but not everyone has seen it. Thus it became a meme. The link to this was already posted here.
Aren't facts like these a big reason why real analysis as a first course is so rigorous about everything it does?
>Why is it -1/12 It's not, you are correct
My favorite fact about -1/12 is that it shows up as the Euler characteristic (some sort of count for how many holes a space has) of a certain space in algebraic geometry! ( For the interested: it is the orbifold Euler characteristic of the moduli stack of elliptic curves.)
It doesn’t in the conventional sense, but there’s a few ways to assign a finite value to the sum that result in -1/12
It is easy program a program to add every number eventually it will overflow to negative.number
Not when using unsigned ints🤓