1+2+3+4... = -1/12

[nickcool9999]

Re: 1+2+3+4... = -1/12

I still don't know why it works, but Numberphile says the value -1/12 for that sum is used for practical quantum physics applications so I have no choice but to believe it.

My theory is black holes.

Well that's a cool idea to be true but unfortunately...

It got myth busted.

YEEEEAAAAA lol

[stargroup100]

Re: 1+2+3+4... = -1/12

There's probably some stupid math algorithm to justify your answer but the fact of the matter is that infinite numbers summed does not produce a negative rational fraction, unless that is you're trying to be completely incorrect...

Wherever it is that you learned that OP, just ignore it. You're making math harder to justify than it needs to be. -.-

I may have fudged the terminology in the opening post for whatever reason but I assure you, I am no stranger to math.

Obviously, for any finite number of terms, the sum of the integers starting from 1 can't possibly be fractional or negative. However, we're talking about an infinite series here, which characteristically makes the problem different.

The reason why this result can even make sense is because we're talking about different contexts. Mathematical structures in different domains have different properties. When you have problems in which multiple answers can be justified, the conditions for the problem must be explained to obtain a particular solution. While in the sense of a divergent series this goes to infinity, through analytic continuation this statement can be justified. The result is actually used in quantum mechanics, and without it we wouldn't have the level of digital communication we have today.

Just because you don't like the idea doesn't make it false. And whether or not pure mathematics is difficult to justify is irrelevant. Fermat's Last Theorem was notoriously hard to justify, but people spent hundreds of years to do it, and it's an important result.

Mathematical Justification or Critical Thinking (Why this board exists):

ƒ(n,s)=1/n^s where (s=-1),
sub as ƒ(n,-1)=1/n^-1
ƒ(n,-1)=1/(1/n)
ƒ(n,-1)=n

Therefore, if (ƒ(n,-1)=-1/12) and ƒ(n,-1)=n, then n=-1/12. So your math is wrong anyways.

ƒ(-1/12,-1)=-1/12
ƒ(ΣI,-1)≠-1/12
∴ΣI≠-1/12

:P

wtf is this

making an assumption with no basis: "Therefore, if (ƒ(n,-1)=-1/12)"
jumping to conclusions: "n=-1/12. So your math is wrong anyways."
another assumption: "ƒ(ΣI,-1)≠-1/12"

clearly math is not your strong suit

[TheSaxRunner05]

Re: 1+2+3+4... = -1/12

Wrecked

[Zapmeister]

Re: 1+2+3+4... = -1/12

i still maintain that it doesn't make sense to talk about a divergent series having a finite value as a sum.

let's go back to my earlier example of

(1-x)^-2 = 1 + 2x + 3x^2 + 4x^3 + 5x^4 + ...

and the expansion is valid for |x|<1

of course everyone agrees that if you plug x=0.1 in here you get 1+0.2+0.03+0.004= ... = 100/81, because x=0.1 is inside the radius of convergence.

you're saying: if i plug in x=10, then i get a value of the sum 1+20+300+4000+ ... = 1/81 which is justified by analytic continuation, since there is a unique analytic continuation of the sum 1+2x+3x^2+4x^3... outside its radius of convergence, and this analytic continuation just happens to be equal to (1-x)^-2.

here's the thing. if you wanted to do that, you'd need to tell me what the function you're trying to take out of its radius of convergence is. and because the series is divergent, you're not going to get the same answers whichever way you do it. so you're trying to say that 1+2+3+4... represents the analytic continuation of

1^-s + 2^-s + 3^-s + 4^-s + ... | s=-1

at s=-1, which is the zeta function at -1, which is -1/12.

there is no way to determine that when you wrote 1+2+3+4... you're starting from the zeta function and working things out from that. you could be trying to get it from the analytic continuation of another function.

here's another way to write it. how about this?

1+2+3+4+... = 1 + 2x + 3x^2 + 4x^3 + ... | x=1
= (1-x)^-2 | x=1
= 0^-2 = "infinity"

you're still not convinced?

1+2+3+4... = 1 + 2x + 3x^2 + 4x^3 + ... | x=1
= (1-x)^-2 | x=1
= ( (1-x)^-1)^2 | x=1
= (1 + x + x^2 + x^3 + ...)^2 | x=1
= (1 + 1 + 1 + 1 + ...)^2
= (zeta(0))^2
= 1/4

yeah. none of this would be happening if you used a valid x inside the radius of convergence, since everything would agree on a certain value.

this is the point. there may be occasions when you're trying to work out what the analytic continuation of 1^-s + 2^-s + 3^-s + ... is at s=-1. in these cases you'd write it as zeta(-1), because the zeta function is defined to be this analytic continuation. you'd never write it out as 1+2+3+4+... because nobody knows what you mean when you're writing down the sum of a divergent series. it's complete trash. just like your face (oohh burn)

[Dynam0]

Re: 1+2+3+4... = -1/12

straight up gangst thread

[Reincarnate]

Re: 1+2+3+4... = -1/12

check out dirichlet series and grandi series

this also reminds me: check out thompson's paradox. edit: http://en.wikipedia.org/wiki/Thomson's_lamp

It's really easy to abuse the concept of infinity

[benguino]

Re: 1+2+3+4... = -1/12

Zapmeister...you can't use (1-x)^(-2) at x=1 as an analytic continuation of 1+2x+3x^2+... because (1-x)^(-2) itself is not analytic at x=1. You have to remember that analytic continuations are unique (in some sense). I still don't see any contradictions or inconsistencies when we use the notion of analytic continuation to "redefine" things.

[stargroup100]

Re: 1+2+3+4... = -1/12

you'd never write it out as 1+2+3+4+... because nobody knows what you mean when you're writing down the sum of a divergent series.

I agree.

The reason why this result can even make sense is because we're talking about different contexts. Mathematical structures in different domains have different properties. When you have problems in which multiple answers can be justified, the conditions for the problem must be explained to obtain a particular solution.

The condition for this problem is talking about the riemann-zeta function (in a nutshell).

[benguino]

Re: 1+2+3+4... = -1/12

I think the best way to explain this is by looking at an analog of a much more simpler example. When it comes to mathematics, we like to define a lot of things. However, sometimes we're stuck by the definition we have which restricts us. So we look for a way to "extend" our definitions to:
1) make everything still consistent with the old definition (i.e. not break maths)
2) have "nice" features in this new definition that is desirable

Let's take an example with exponentiation. When first introduced to exponentiation, we learn this as repeated multiplication: if we see something like 2^3, it's simple: 2^3 = 2*2*2 = 8. However, then you see something like 4^(3/2) and you're like "da fuq? how am I supposed to do one and a half multiplications?" With our original definition, we are powerless, we can not do anything, we are dumbfounded with the notion of trying to 1.5 multiplications. However, we can redefine exponentiation for fractional powers in a way that is still consistent with the previous rules and we find that 4^(3/2) = sqrt(4)^3 = 2^3 = 2*2*2 = 8.

We had to redefine what exponentiation meant for fractional powers to solve the problem. Did we redefine in any ol' way? No, we had to redefine it in a way that was still consistent with the previous rules. Think of this as the "fractional continuation" of the exponentiation function. We could have left it "as is" and just say it means nothing when our power isn't a whole number, but we wanted to work beyond the whole numbers. Mathematicians have extended this definition all the way up to the real numbers.

What about complex numbers? Well, looking only at the definition for what we have for real numbers, we can't do anything. We can redefine exponentiation for complex numbers quite nicely in a way that makes the function f(z) = z^c analytic, where z and c are complex numbers. What it means for a function to be analytic is beyond the scope of this post, but it's basically a "nice" property that is desirable for a function with complex variables. Although technically, this isn't an analytic continuation of f from the reals to the complex, the idea is nearly the same: we can extend the domain of many complex functions using analytic continuation (and since it can be shown that analytic continuation is unique, we have less to worry about in terms of inconsistencies!)

So although 1+2+3+4... doesn't really equal anything (unless you count infinity) with our original definition (i.e. the value of the limit of the partial sums), we can extend our definition by means of analytic continuation and result in -1/12.

So is 1+2+3+4...=-1/12? Both yes and no. Sticking with the usual definition which most of us are familiar with, this is bogus nonsense. Using this other definition which some mathematicians (particularly complex analysts and it seems quantum physicists use): yes. In this re-definition correct? Definitions are neither correct or incorrect, they just are so whether this re-definition using analytic continuation is a good one...that's your call.

[igotrhythm]

Re: 1+2+3+4... = -1/12

barf

Excuse me while I pick up the pieces of my brain that are all over the floor.

[Riotpolice]

Re: 1+2+3+4... = -1/12

This thread has got to be the flagship of the critical thinking thread.

[stargroup100]

Re: 1+2+3+4... = -1/12

which is funny because there's really not much debate in this thread

[Zapmeister]

Re: 1+2+3+4... = -1/12

Zapmeister...you can't use (1-x)^(-2) at x=1 as an analytic continuation of 1+2x+3x^2+... because (1-x)^(-2) itself is not analytic at x=1.

crap... you got me. you got me damn good.

best i can say now is that if you try to define sums of divergent series using the formulae and stuff you derive from convergent series, you have to throw away a lot of favourable nice assumptions you make about how they behave.

e.g.

sum(n: n=1 to infinity) = x (here x is assumed to be -1/12 but that doesn't matter)
=> sum(n: n=0 to infinity) = x
=> sum(n-1: n=1 to infinity) = x
=> sum(n: n=0 to infinity) - sum(n-1: n=1 to infinity) = 0
=> sum(1: n=1 to infinity) = 0
=> sum(1: n=0 to infinity) = 0
=> sum(1: n=0 to infinity) - sum(1: n=1 to infinity) = 0
=> 1=0

however i'm pretty sure term-by-term addition/subtraction like this is not justified in the way stargroup100 is trying to treat these things, although he never explicitly states that in any of his posts.

i'm still trying to find a contradiction that i'm satisfied with but for the moment it looks like you got me here >_<

edit: i was hoping my 300th post would be less self-demeaning than this but whatever

[rushyrulz]

Re: 1+2+3+4... = -1/12

I'd have to say, the most intriguing mathematical truth is:

[stargroup100]

Re: 1+2+3+4... = -1/12

however i'm pretty sure term-by-term addition/subtraction like this is not justified in the way stargroup100 is trying to treat these things, although he never explicitly states that in any of his posts.

you're right, I didn't explicitly state anything of the sort. because I agree with you.