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Lionhart

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Joined: 2015-02-17

Posts: 40

1+2+3+4+5+... = -1/12 ???

[Disclaimer: if you watch Numberphile this is old news.]

I learned this pretty interesting fact today: the infinite sum of all natural numbers converges to -1/12.

Non rigorous proof:
Series G = 1-1+1-1+1-1+...
Well, what is G equal to? Most people would say there is no answer. It just toggles back and forth between 1 and 0. This mentality is fine if you enjoy living in the dark, but in actuality the answer is 1/2. Take a look at this:

What is 1-G? It's 1-(1-1+1-1+1-1+...). If we distribute the -1, we end up with
1-G = 1-1+1-1+1-1+...

Well, wait, isn't that just G? That means 1-G=G. Add G to both sides. 1=2G. Now divide by 2. 1/2=G.

Boom. Ok, moving right along.

Series R = 1-2+3-4+5-6+7-...

Let's find 2R. Except, I'm going to shift it one place to the right.

 1-2+3-4+5-6+7-...
+0+1-2+3-4+5-6+7-...
___________________  
 1-1+1-1+1-1+1-1+...

Interesting. So now we know that 2R=G.

G is still 1/2, so R is actually 1/4.

If your mind hasn't started to decompose, let's keep moving.

Let's take the original series, 1+2+3+4+5+6+7+... (hereon referred to as S) and subtract R.

  1+2+3+4+5+6+7-...
-(1-2+3-4+5-6+7-...)
___________________  
0+4+0+8+0+12+0...

Interesting. Now let's factor out a 4.
4(1+2+3+4+5+6+7+...) = S-R
Now let's replace for S then combine them.

4S=S-R
3S=-R
R still equals 1/4, so...
3S=-1/4
Therefore,
S=-1/12.

This isn't mathematical trickery like proving π=4 or something. This is Riemann's Zeta function evaluated at -1. This answer is used in String Theory textbooks, like Joseph Polchinski's String Theory Vol. 1 (Page 22). And it works when you replace it in formulas.

Amazing!

Comments?

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Ratburntro44

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Re: 1+2+3+4+5+... = -1/12 ???

This isn't mathematical trickery like proving π=4 or something.

You're right, it isn't mathematical trickery, it's verbal trickery. The only thing thing that makes this impressive is using the unqualified term "sum" to refer to a specific type of sum that it would not be normally used to mean and then using extraneous proofs to demonstrate things that are only actually true in a specific context.

For an infinite series, the manipulations that you make in your proofs rely on steps that are only valid for finite summations. For example, using steps similar to what you have given:
S=1+2+3+4+…=0+1+2+3+4+…
0=S-S=1+1+1+1+1+…=0+1+1+1+1+1
Taking the difference between 1+1+1+1+…=0 and 0+1+1+1+1+…=0 gives you 1=0, which is clearly incorrect, demonstrating that the non-rigorous methods used to demonstrate this sum are not at all a good basis for claiming that the sum of the natural numbers is -1/12.

That is not to say, however, that there are not a variety of non-standard summation methods that do yield the result of -1/12, matching what is seen through the non-rigorous method you demonstrate. While in a normal context it is not really correct to simply refer to these as the "sum", it is true that they have important application in physics and clearly do have a meaning. However, my point still stands; the main reason this "interesting fact" seems so interesting is the deliberate abuse of terminology combined with non-rigorous methods to misrepresent mathematics in order to sell it as something cool.

Lionhart

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Joined: 2015-02-17

Posts: 40

Re: 1+2+3+4+5+... = -1/12 ???

This isn't mathematical trickery like proving π=4 or something.

You're right, it isn't mathematical trickery, it's verbal trickery. The only thing thing that makes this impressive is using the unqualified term "sum" to refer to a specific type of sum that it would not be normally used to mean and then using extraneous proofs to demonstrate things that are only actually true in a specific context.

For an infinite series, the manipulations that you make in your proofs rely on steps that are only valid for finite summations. For example, using steps similar to what you have given:
S=1+2+3+4+…=0+1+2+3+4+…
0=S-S=1+1+1+1+1+…=0+1+1+1+1+1
Taking the difference between 1+1+1+1+…=0 and 0+1+1+1+1+…=0 gives you 1=0, which is clearly incorrect, demonstrating that the non-rigorous methods used to demonstrate this sum are not at all a good basis for claiming that the sum of the natural numbers is -1/12.

That is not to say, however, that there are not a variety of non-standard summation methods that do yield the result of -1/12, matching what is seen through the non-rigorous method you demonstrate. While in a normal context it is not really correct to simply refer to these as the "sum", it is true that they have important application in physics and clearly do have a meaning. However, my point still stands; the main reason this "interesting fact" seems so interesting is the deliberate abuse of terminology combined with non-rigorous methods to misrepresent mathematics in order to sell it as something cool.

You can't possibly expect a thorough proof on analytic continuation on this forum. This proof is the exact one given by Dr. Tony Padilla, a physicist at the University of Nottingham. It's also the only one that I, like the members of this forum, have the patience to learn. Is it rigorous? No, but that does not make it the trash that you are describing, only approachable given our audience. Is it vague? Sure. If people want to learn the specifics, they may.

And, after reading your counterexample, I believe that it is vague instead of wrong because it is a simplified version of the following proof
http://www.nottingham.ac.uk/~ppzap4/response.html

In summation (pun totally intended), being vague is not a crime, especially when dealing with non-maths-majors.

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Twipply

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Joined: 2015-02-28

Posts: 32

Re: 1+2+3+4+5+... = -1/12 ???

The only thing thing that makes this impressive is using the unqualified term "sum" to refer to a specific type of sum that it would not be normally used to mean and then using extraneous proofs to demonstrate things that are only actually true in a specific context.

Waow.

Is it rigorous? No, but that does not make it the trash that you are describing

Sry bro it's trash.

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Anch

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Joined: 2015-02-16

Posts: 5,447

Re: 1+2+3+4+5+... = -1/12 ???

I see.

skullz17

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Joined: 2015-02-15

Posts: 6,699

Re: 1+2+3+4+5+... = -1/12 ???

I see.

Don't lie.

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SPT

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Joined: 2015-02-17

Posts: 751

Re: 1+2+3+4+5+... = -1/12 ???

i can see with my eyes too

Anch

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Joined: 2015-02-16

Posts: 5,447

Re: 1+2+3+4+5+... = -1/12 ???

I see.

Don't lie.

I hear.

Jabatheblob1

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Joined: 2015-03-01

Posts: 856

Re: 1+2+3+4+5+... = -1/12 ???

This is all a theory and actually doesn't work. My friend has this conversation with my math professor every class and she has brought out books on books pointing out why this doesn't work. I can find the books tomorrow for you if you would like to know why these can't possibly work.

One thing that i remember is that you can't set anything equal to a series that never ends. Because to get the result of something that never ends is nonsense. That's like asking for the last number of infinity. You can't set a number value to a never ending theory.

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Ratburntro44

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Joined: 1970-01-01

Posts: 1,383

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Re: 1+2+3+4+5+... = -1/12 ???

This is all a theory and actually doesn't work. My friend has this conversation with my math professor every class and she has brought out books on books pointing out why this doesn't work. I can find the books tomorrow for you if you would like to know why these can't possibly work.

One thing that i remember is that you can't set anything equal to a series that never ends. Because to get the result of something that never ends is nonsense. That's like asking for the last number of infinity. You can't set a number value to a never ending theory.

That's… not really how it works.

Of course under the standard definition for summation of infinite series (limit of the partial sums) it does not have any value. The forums of summation in which this is true have different definitions which allow this to be the case. In the case of the 1-1+1-1+1-1+1-1+…, you can get a value from the Cesàro sum, which is defined as the limit of the mean of the partial sums. In this case, the means of the partial sums would be 1,0.5,0.67,0.5,0.6,0.5,0.57,0.5,…, which easily be shown to converge to 0.5. This definition is useful because when the standard summation exists, it is always equal to the standard sum; this means than it is a generalization, rather than simply a replacement.

In the case of 1+2+3+4+…, the Cesàro sum does not have a value either. However, there is another generalization of summation that does have a value here; namely, the Ramanujan sum. It is not quite as simple of a definition as the Cesàro sum, but the notable part is that under this type of summation 1+2+3+4+… does sum to -1/12. In addition to this, there is another method called zeta function regularization that gives the same result. Basically, this uses the Riemann zeta function, which extends the function for the sums of infinite series of the form 1/n^s in a way that allows you to assign a "sum" which is consistent with the traditional sum when it converges and assigns a new value (again -1/12 in this case) to what would otherwise be divergent.

The main reason people argue that this doesn't work is as a result of people (like OP) trying to pass it off as normal summation, even though it very clearly isn't. At the same time, it is not in any way wrong; as is pretty much always the case, you can get different answers when you use different definitions. This is no exception. One of the biggest things mathematicians seek to do is generalize definitions and theorems to apply in cases where they were not seen before; this is merely an example of the logical result of generalizations of summation that allow divergent sums to be assigned a value. It is only nonsense when you use definitions (such as the standard definition) that leave it undefined.

Altogether, however, this does back up my point to OP that it is counterproductive to tote this "fun fact" as something other than what it actually is- that is to say, a curious and sometimes useful result that you get from using non-standard definitions.

Jabatheblob1

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Joined: 2015-03-01

Posts: 856

Re: 1+2+3+4+5+... = -1/12 ???

So... you're finding the answer to a series rather than the theory of an infinite repeating function. Difference being that a series has a beginning and an end. The theory of infinity doesn't have an end. So you can't find the summation of a function that has no evident end. This is the problem. A theory cannot be a number. I agree with what you're saying because all your arguments argue for that it is a convergent series. Meaning it is a set of numbers... now if you put this same argument to a non-convergent or a divergent series then i would have to disagree with you. Cesàro sum even talks about how it is the summation of a convergent series rather than divergent. Essentially we are arguing two different points.

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Sybil

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Joined: 2015-02-15

Posts: 171

Re: 1+2+3+4+5+... = -1/12 ???

sorry, google says 1+1 = 2

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Bimps

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Joined: 2015-02-08

Posts: 5,067

Re: 1+2+3+4+5+... = -1/12 ???

uh i can disprove you since i know math
lets look at the title
1+2+3+4+5+... = -1/12 ???
first of all that is not right, dots arent in math they are in grammar
why is there a + after the 5 that means syntax error
wt are those ?'s doing there wt is wrong with you

thank you this has been math with bimps

Ratburntro44

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Joined: 1970-01-01

Posts: 1,383

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Re: 1+2+3+4+5+... = -1/12 ???

So... you're finding the answer to a series rather than the theory of an infinite repeating function.

That doesn't even make sense.

Difference being that a series has a beginning and an end.

That is patently untrue.

The theory of infinity doesn't have an end.

I'm not totally sure what you mean by "theory of infinity" here, but I guess I can agree that infinite things, by definition, do not have an end.

So you can't find the summation of a function that has no evident end.

That is patently untrue.

This is the problem. A theory cannot be a number.

I have no idea what you are trying to say by this.

I agree with what you're saying because all your arguments argue for that it is a convergent series.

I'm not claiming that what are presented in OP are convergent series, I am claiming that they are divergent series which, under abnormal definitions of summation, can be given sums.

Meaning it is a set of numbers…

Not really.

now if you put this same argument to a non-convergent or a divergent series then i would have to disagree with you.

I'm not really totally sure what argument you think I'm making at this point, but I can assure you that under these nonstandard definitions for summation divergent series can be assigned sums (this is the whole point of making nonstandard definitions).

Cesàro sum even talks about how it is the summation of a convergent series rather than divergent.

I don't know what you're referring to here, but the Cesàro sum is perfectly capable of assigning sums to both all convergent series and some divergent series.

Essentially we are arguing two different points.

That may be true, but I have a hard time telling what point you are even attempting to make here, since a lot of what you say doesn't really seem coherent.

Bimps

Member

Joined: 2015-02-08

Posts: 5,067

Re: 1+2+3+4+5+... = -1/12 ???

So... you're finding the answer to a series rather than the theory of an infinite repeating function.

That doesn't even make sense.
you dont make sense

Difference being that a series has a beginning and an end.

That is patently untrue.
your patently untrue

The theory of infinity doesn't have an end.

I'm not totally sure what you mean by "theory of infinity" here, but I guess I can agree that infinite things, by definition, do not have an end.
infinity has an end it ends with y

So you can't find the summation of a function that has no evident end.

That is patently untrue.
oh so now you are repeating yourself you ****

This is the problem. A theory cannot be a number.

I have no idea what you are trying to say by this.
your patently untrue

I agree with what you're saying because all your arguments argue for that it is a convergent series.

I'm not claiming that what are presented in OP are convergent series, I am claiming that they are divergent series which, under abnormal definitions of summation, can be given sums.
thats a lot of words

Meaning it is a set of numbers…

Not really.
you are wrong

now if you put this same argument to a non-convergent or a divergent series then i would have to disagree with you.

I'm not really totally sure what argument you think I'm making at this point, but I can assure you that under these nonstandard definitions for summation divergent series can be assigned sums (this is the whole point of making nonstandard definitions).
i cant read

Cesàro sum even talks about how it is the summation of a convergent series rather than divergent.

I don't know what you're referring to here, but the Cesàro sum is perfectly capable of assigning sums to both all convergent series and some divergent series.
i cannot speak italian

Essentially we are arguing two different points.

That may be true, but I have a hard time telling what point you are even attempting to make here, since a lot of what you say doesn't really seem coherent.
dont call him not smart that isnt nice

thank mr skeltal

Bobithan

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Joined: 2015-02-15

Posts: 4,476

Re: 1+2+3+4+5+... = -1/12 ???

thank

What are you trying to prove here? You didn't even thank correctly. When thanking somebody, you say "Thank you", not just "thank". I'm sorry if English isn't your first language, but this is unforgivable.

mr

What is Rat is not actually a Mr but a Ms? Be considerate.

skeltal

Gibberish. Your argument is invalid.

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Bimps

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Joined: 2015-02-08

Posts: 5,067

Re: 1+2+3+4+5+... = -1/12 ???

AHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH

FDOOU

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Re: 1+2+3+4+5+... = -1/12 ???

Dude that's not even how you say that it's only As where did these Hs come from?

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Bimps

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Joined: 2015-02-08

Posts: 5,067

Re: 1+2+3+4+5+... = -1/12 ???

the h store

Freckleface

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Posts: 1,364

Re: 1+2+3+4+5+... = -1/12 ???

If thats the name of the store then you forgot to capitalize the first letters: The H Store
If not then please tell wat it is called because i ave been lacking many of my Hs in my speec lately :/

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F

Bimps

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Joined: 2015-02-08

Posts: 5,067

Re: 1+2+3+4+5+... = -1/12 ???

If thats the name of the store then you forgot to capitalize the first letters: The H Store
If not then please tell wat it is called because i ave been lacking many of my Hs in my speec lately :/

hint: you arent funny

Jabatheblob1

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Joined: 2015-03-01

Posts: 856

Re: 1+2+3+4+5+... = -1/12 ???

I asked my math professor and she said this is absolute nonsense...

Definition of infinity by math terms: "Infinity is the idea of something that has no end."

In math. You cannot find an answer to something that goes on infinity. My teacher said that there are things called sets. Which may be what you're talking about.

What i am saying is if you have a function

int i = 0;
while(1=1)
{
    i++;
}
i--;
console.write(i.tostring());

You're trying to find out what i is. A never ending function doesn't have an end value. That's why infinity is an idea rather than a number.

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Fradeshan

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From: nyc

Joined: 2015-02-15

Posts: 366

Re: 1+2+3+4+5+... = -1/12 ???

itt

[words]

I learned this pretty interesting fact today/last week/last month ...

[words]

One thing that i remember from school/class/a youtube video is that...

[words]
[words that make sense]
[words]

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iPwner

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From: CaliforNYAN Land.

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Re: 1+2+3+4+5+... = -1/12 ???

Nice numbers! I believe it is alternatively called  "Paquito's triangle", originally had something to do with the shape of beannated chicken tacos in a 1950's barnyard... the more you know ! !

Also, I'd like to say that in case you were wondering, I am the one in posession of the account "lionheart"... I just forgot the pass and email. ;D It's interesting to see someone who wanted to have that exact same name on the forums. :o , But due to a decision I made 3 years ago, he was unable to create the correct spelling of that account!

Flattering.

Ah, ah.

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Bimps

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Re: 1+2+3+4+5+... = -1/12 ???

what math