How Does Summing Infinite Integers Equal Negative One-Twelfth?

[zachx]

I've recently accepted (reluctantly) that 1+2+3+4+...=-1/12, but wouldn't that mean the limit as x→∞ of f(x)=x not be ∞ but some maybe negative number because if it were ∞ then 1+2+3+4+... should also be ∞. Also, do numbers get so positive they become negative lol.

 

[Vanadium 50]

I've recently accepted (reluctantly) that 1+2+3+4+...=-1/12

Well you shouldn't. That series is divergent.

 

[certainly]

Well you shouldn't. That series is divergent.

It's actually quite useful in the study of divergent series and is called Ramanujan Summation.

 

[zachx]

Well you shouldn't. That series is divergent.

I thought that too, but this guy pulled out some string theory textbook.

 

[zachx]

It's actually quite useful in the study of divergent series and is called Ramanujan Summation.

Oh, never mind thanks for clearing it up for me, but I still don't quiet understand why it's negative n

 

[certainly]

Oh, never mind thanks for clearing it up for me, but I still don't quiet understand why it's negative n

Is that why you've been calling yourself a cat :-)
In regards to your question, it's a different type of summation, so I don't know how to make sense of it. (i.e. in the traditional sense the sum is still divergent).
I found an old thread that has a post by yasiru89 with a more formal derivation of it, may help.

 

[zachx]

Is that why you've been calling yourself a cat :)

Moo.

Alright, well 1 last question: so when you multiply, you're adding a bunch of times and when you integrate you're finding the area, so conceptually what are you doing with the Ramanujan Summation?

 

[certainly]

so when you multiply, you're adding a bunch of times and when you integrate you're finding the area, so conceptually what are you doing with the Ramanujan Summation?

I don't think a geometrical interpretation exists. (if anyone knows of any, kindly point them out). However several analytical proofs exist.
[For instance you could use the relation between the zeta function and the Dirichlet eta function $(1-2^{1-s})\zeta (s)=\eta (s)$ and then Abel sum $\eta (-1)$.]

 

[muzialis]

I think the "geometrical interpretation" is the fact that the Ramahujan summation gives the first term in an asymptotic series related to the divergent sum. The matter is explained best here, https://terrytao.wordpress.com/2010...tion-and-real-variable-analytic-continuation/.
Once this is understood, even the physical application of the divergent sum regularisation (for example, in estimating the Casimir effect) make some sense.
There are other cases where divergent sums are useful in practice: for example Stirling's approximation of the factorial. It is given by a divergent series, but, fixing the number of terms in the series, the approximation for very large argument is very good (in a sense made precise by the definition of asymptotic series).

 

[micromass]

I thought that too, but this guy pulled out some string theory textbook.

God, I hate that video. The video is very misleading. I hoped they would be somewhat clear in it.

First of all, the series $1+2+3+4+...$ diverges. You will find no mathematician that disagrees with this. The most natural sum is $1+2+3+4+... = +\infty$.
Now, what is the $-1/12$ thing all about? Well, some mathematicians have found a way to associate a number to divergent series. I would not call that number the "sum" of the series, it is just a number associated to it. In this case, the number associated to $1+2+3+4+...$ is $-1/12$. Now, we often write $1+2+3+4+5+... = -1/12$, but that's where you should be careful, since that $=$ sign does not mean the classical one, in fact it means that we evaluate the series in a nonstandard way (like Ramanujan summation). Now in many circumstances, replacing $1+2+3+4+...$ with $-1/12$ is wrong and a very bad idea, but in some it might work out. It should then be shown why exactly we can replace the sum by $-1/12$.

 

[Drakkith]

God, I hate that video. The video is very misleading. I hoped they would be somewhat clear in it.

First of all, the series 1+2+3+4+...1+2+3+4+... diverges.

Oh thank god. That video was driving me crazy. I was telling myself, "There's no way that's right!".

 

[micromass]

Oh thank god. That video was driving me crazy. I was telling myself, "There's no way that's right!".

That video is the equivalent of a teacher telling a high school class that there is a number whose square is $-1$, without saying that it is not a real number and without saying that we are essentially inventing a new number system. Sure, it will get you likes because it is mindblowing, but it is pedagogically awful.

 

[Vanadium 50]

I agree with micromass, the problem is the equals sign, because it doesn't mean, well, "equals". Maybe a different symbol, like a double-ended arrow, would be less confusing. Or a functional that took a series as input.

 

[phion]

Made me grin like an idiot.

 

[wabbit]

This is an interesting interpretation of some divergent sumations : http://math.ucr.edu/home/baez/counting/

 

[hunt_mat]

It's to do with the analytic continuation of the Riemann zeta function. The things they do in that video are slight of hand (typical physicists) and so you shouldn't take their derivation seriously at all.

I was taught by Ed Copeland at university.

 

[davidmoore63@y]

I've always thought the easier example is
X= 1+2+4+8+16+32+..
2x= 2+4+8+16+32+...
Subtract
-x= 1
X=-1
Nice

 

[Mark44]

I've always thought the easier example is
X= 1+2+4+8+16+32+..
2x= 2+4+8+16+32+...
Subtract
-x= 1
X=-1
Nice

The main problem here is that you are doing arithmetic with two series that obviously diverge. In effect the arithmetic is leading one to believe that the indeterminate form [∞ - ∞] gives a meaningful answer.

 

[micromass]

The main problem here is that you are doing arithmetic with two series that obviously diverge. In effect the arithmetic is leading one to believe that the indeterminate form [∞ - ∞] gives a meaningful answer.

But in fact, if the series converge, then the answer is correct. That is a big if of course, since in the usual real numbers, it doesn't converge. But in the 2-adic numbers, it does! And there we indeed have that $-1 = 1 +2 + 4 + 8 + ...$. But this is a completely different thing that the standard reals!

 

[thelema418]

It's to do with the analytic continuation of the Riemann zeta function. The things they do in that video are slight of hand (typical physicists) and so you shouldn't take their derivation seriously at all.

I was taught by Ed Copeland at university.

Is there a better video available on that topic? I am curious because high school teachers have told me that they use that video in their classroom.

 

[NotGauss]

You cannot use the standard properties of mathematics, such as the commutative or associative properties on infinite sums, they break down. Bad math.

 

[JonnyG]

That video made me want to blow my brains out.

 

[cuallito]

I have a Bachelor's in Physics.

My 2 cents: It's understood to mean -1/12 + Infinity in physics, but since infinities are bad, physicists just throw them out and use the -1/12 to do calculations.

 

[FactChecker]

These "summations" can only be used in restricted ways. For instance, you can not insert terms of 0 into the summation without changing the results.
Similarly, this theory says that 1 - 1 + 1 - 1 + 1 ... = 1/2 because the partial summations are 0 half the time and 1 the other half time. But they never come close to 1/2. I can understand how this could be useful in studying random behavior, where the expected value of partial sums of random lengths might be 1/2. I wonder if the real value of this theory is in applications to random processes. The significance of inserting terms of 0 would then be easy to appreciate.

The original summation is still hard to accept. Is the value -1/12 unique, or can you use similar logic to get other answers? A "proof" would have to show that -1/12 is the only possible answer. The "proof" in the link does not show that. In fact, the "proof" in the link starts with the assumption that 1-1+1-1+1... = 1/2. I think that the -1/12 does more to prove that the whole thing is wrong than anything else. If they assumed in the link that 1-1+1-1+1... was anything other than 1/2, that would change -1/12, wouldn't it?

 

[WWGD]

Actually, even if this sum converged in a standard sense (which it doesn't) , the = sign does not denote equality, but only convergence, and this depends on a choice of topology. So, like Fact Checker said, you need a reasonable notion of convergence to use the = sign. Or , like others said, an explanation for what the = means. A sort of Clintonian description of "What 'is' is".

 

[WWGD]

Is there a better video available on that topic? I am curious because high school teachers have told me that they use that video in their classroom.

I'll look for a video if you tell me the name of the high schools where they teach this garbage so I can tell everyone to avoid those schools.

 

[FactChecker]

In regards to your question, it's a different type of summation, so I don't know how to make sense of it. (i.e. in the traditional sense the sum is still divergent).
I found an old thread that has a post by yasiru89 with a more formal derivation of it, may help.

Romanughan was certainly a genius. There must be some use for this. But it would take some real studying of formal proofs to understand how to use this. Your link looks like it might be rigorous enough. Other links and "proofs" I have seen leave a lot to be desired.

 

[atyy]

God, I hate that video. The video is very misleading. I hoped they would be somewhat clear in it.

First of all, the series $1+2+3+4+...$ diverges. You will find no mathematician that disagrees with this. The most natural sum is $1+2+3+4+... = +\infty$.
Now, what is the $-1/12$ thing all about? Well, some mathematicians have found a way to associate a number to divergent series. I would not call that number the "sum" of the series, it is just a number associated to it. In this case, the number associated to $1+2+3+4+...$ is $-1/12$. Now, we often write $1+2+3+4+5+... = -1/12$, but that's where you should be careful, since that $=$ sign does not mean the classical one, in fact it means that we evaluate the series in a nonstandard way (like Ramanujan summation). Now in many circumstances, replacing $1+2+3+4+...$ with $-1/12$ is wrong and a very bad idea, but in some it might work out. It should then be shown why exactly we can replace the sum by $-1/12$.

How about this presentation? Is it any better?

http://math.ucr.edu/home/baez/numbers/24.pdf

 

[thelema418]

I'll look for a video if you tell me the name of the high schools where they teach this garbage so I can tell everyone to avoid those schools.

I wouldn't even say that they "teach" it, as it is not part of the curriculum map at any high school. Usually teachers show these types of resources because a student, at some point in time, shared the resource with them. Since it is a resource that keeps students involved or interested in mathematics, they show it to future classes. One aspect of mathematics pedagogy is to foster inquiry, wonderment, or what Haddamard and Poincaré called creativity in mathematics.

Personally, I'm now looking at this video as an opportunity for students to challenge some of the ideas presented. What may be hard for students (and the teacher) is that the video depicts "mathematical experts." Quite honestly, how can the standard middle school or high school teacher challenge the experts unless other resources are available with better articulation for the general public?

 

[WWGD]

I wouldn't even say that they "teach" it, as it is not part of the curriculum map at any high school. Usually teachers show these types of resources because a student, at some point in time, shared the resource with them. Since it is a resource that keeps students involved or interested in mathematics, they show it to future classes. One aspect of mathematics pedagogy is to foster inquiry, wonderment, or what Haddamard and Poincaré called creativity in mathematics.

Personally, I'm now looking at this video as an opportunity for students to challenge some of the ideas presented. What may be hard for students (and the teacher) is that the video depicts "mathematical experts." Quite honestly, how can the standard middle school or high school teacher challenge the experts unless other resources are available with better articulation for the general public?

I agree that one should encourage the challenge of ideas, but it seems like a lot of background is needed to be able to present this challenge in a coherent way; maybe if it is a really advanced class, but do you think you can use concepts like analytic continuation, at the high school level?

Maybe if you have someone particularly good at presenting the general ideas of convergence, continuation, but this is rare. Sorry if my post came off as aggressive.

 

[atyy]

I agree that one should encourage the challenge of ideas, but it seems like a lot of background is needed to be able to present this challenge in a coherent way; maybe if it is a really advanced class, but do you think you can use concepts like analytic continuation, at the high school level?

Maybe if you have someone particularly good at presenting the general ideas of convergence, continuation, but this is rare. Sorry if my post came off as aggressive.

But historically the formula was discovered without analytic continuation. Is it so terrible to present it analogous to the way old quantum theory is still taught before quantum mechanics? I'm pretty sure all students, even the worst, know that the "=" sign is not the usual one.

 

[atyy]

Here is another interesting comment on the formula and the place of "formal derivations" by Atiyah in his essay "How Research is Carried Out" (https://books.google.com/books?id=YJ0cZwxLECAC&printsec=frontcover#v=onepage&q&f=false, p213)

"The next point I want to mention is the distinction between formalism and rigour. Again this is a dichotomy in mathematics which has a long history. In the formal kind of mathematics you do things without worrying too much about precisely what they mean as long as they give the right answer. You might say this only happens in applied mathematics but that is not quite correct; I think that it also occurs in pure mathematics. Of course, historically, there are famous examples of the formal approach. The most famous practitioner, no doubt, was Euler, who, as you know, produced many magnificent formulae. He "evaluated" such wildly divergent series like $\sum_{n=1}^{\infty} n = -1/12$. It was only a century or so later that precise meaning could be attached to such formulae."

 

[Michael David]

At about 1:11:20 ish he starts talking about 1+2+3+...

I found this quite helpful