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October 30th, 2015, 03:19 PM   #1
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Question The relationship between 1+2+3+4+... and -1/12

I have seen a few videos on YouTube that seem to be saying that -1/12 has something mysterious to do with infinity. In one video a maths professor talks about it for over 15 minutes where he claims it is the nugget of gold that remains when the dirt of infinity is removed, and he claims it is like magic.

In another video called "ASTOUNDING: 1 + 2 + 3 + 4 + 5 + ... = -1/12" they claim to prove that the sum of all natural numbers equals -1/12.

Intrigued by all this, and not being a believer in 'infinity', I decided to do my own investigation into where this -1/12 value comes from. I presented my findings in my own 'response video' here:



In this video I make some outrageous claims, but this does appear to be the most obvious answer for someone like me, who does not believe in the 'magic' of infinity.

But I am only a Systems Analyst (or computer programmer if you prefer) and so I would be very interested in a mathematician's critique of my findings.

Many thanks in advance.
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October 30th, 2015, 03:53 PM   #2
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Originally Posted by Karma Peny View Post
I have seen a few videos on YouTube that seem to be saying that -1/12 has something mysterious to do with infinity. In one video a maths professor talks about it for over 15 minutes where he claims it is the nugget of gold that remains when the dirt of infinity is removed, and he claims it is like magic.

In another video called "ASTOUNDING: 1 + 2 + 3 + 4 + 5 + ... = -1/12" they claim to prove that the sum of all natural numbers equals -1/12.

Intrigued by all this, and not being a believer in 'infinity', I decided to do my own investigation into where this -1/12 value comes from. I presented my findings in my own 'response video' here:



In this video I make some outrageous claims, but this does appear to be the most obvious answer for someone like me, who does not believe in the 'magic' of infinity.

But I am only a Systems Analyst (or computer programmer if you prefer) and so I would be very interested in a mathematician's critique of my findings.

Many thanks in advance.
I watched the video which I thought was interesting, though lacking in any kind of rigor, but it didn't actually address one important thing.

What we are talking about here is the zeta function, which for whole numbers is defined as $\displaystyle \zeta (s) = \sum_{k = 1}^{\infty} \frac{1}{k^s}$. (Actually, this works for all complex s with a real part greater than 1, but this does not affect my argument.)

Note that we are not defining the zeta function for s = -1 by this series. The value of $\displaystyle \zeta (-1)$ is found by a process called analytic continuation and has a representation which has nothing to do with the infinite sum of whole numbers. QFT and String Theory uses 1 + 2 + 3 + ... = -1/12 but this is to be read with a grain of salt: Physics needs to calculate a divergent sum and uses a method to get the answer that it needs. No one is claiming that the method is comfortable...it simply gives the correct answer to the overall calculation. But I want to make it clear: 1 + 2 + 3 + ... is not -1/12.

It would have been nice if the video had mentioned all this but the authors clearly had their own agenda.

-Dan

PS Here is the Wikipedia article on the Reimann-Zeta function. So far as I know it is accurate. Partway down the page is a link on analytic continuation. It will do for an introduction to the subject.
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October 30th, 2015, 04:13 PM   #3
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I think that your core complaint is that those making the (apparently outrageous) claim are playing fast and loose with convergence. Well, yes they are! In fact, as any good precalc student knows, the series 1 + 2 + 3 ... diverges.

BUT... that's not the end of the story! There is a formal sense in which the series converges to -1/12. But it could only make sense if the relevant error terms were becoming negligible, and they aren't. But this itself depends on the metric we use. With the usual (Euclidean/Archimedian) metric, the partial sums diverge. But there are other metrics with respect to which the partial sums converge! Further, there are other types of summation which generalize the usual summation, in the following sense: for any series which converges in the usual sense, it converges in the new sense to the same value, but some series which diverge in the usual sense converge (to a finite value) in the new sense!

Without question, the series 1 + 2 + 3 + ... diverges when we're using the normal meaning of summing an infinite series and the usual metric. But there are other senses in which it has a new value, and as far as I know all of the ways in which it converges shows it converging to the same value, -1/12.

At this point you might say that this is well and good, that mathematicians can play and game that they like and define funny things into existence. (Yes, they can! ) But even if your faith is shaken a bit by seeing that Caesaro summation and Ramanujan summation and all kinds of others agree that the answer 'should' be -1/12, you might still hold out that these don't have real applications. But that turns out to be very, very wrong.

It turns out that when trying to understand the equations that seem to govern the universe, at least at small scales, quantum physicists often encounter (apparently) infinite quantities for sums that appear as solutions to real-world problems. This is problematic, because we don't actually see (for example) infinite energy, or infinite mass, and we don't think that it would be possible. But it turns out that through a collection of mathematical 'tricks' called renormalization, physicists can transform these apparent infinities into numbers like -1/12... and when they do, their predictions match our universe! It's like reality is somehow telling us that there are senses in which sums like 1 + 2 + 3 + ... have finite, negative values.

But there's more! Perhaps you don't believe in empirical evidence, or quantum mechanics, despite its success not only in predicting behavior but in enabling technologies like high-speed computer processors and RAM; medical technologies like MRI/NMR and PET scans, imaging like the scanning tunneling microscope, ultra-precise clocks enabling GPS, lasers and their endless applications, etc. You might still find the elaborate and extremely beautiful concept of analytic continuation convincing. In particular, for large classes of functions (called analytic functions, naturally), learning the values of the function in a disc -- even a tiny disc -- allows us to compute all the values of the function anywhere in $\mathbb{C}$. By this same process, we can extend the function
$$\sum_{k=1}^\infty\frac{1}{k^s}$$
to every complex number $s\ne1.$ In particular, by choosing $s=-1$ you get the sum 1+ 2 + 3 + ..., and the analytic continuation gives us... you guessed it... -1/12.

So this is a long and deep rabbit hole to go down, and I hope I didn't lose you. Your point is quite correct in a certain sense, but it turns out that reality seems to often want to use a different sense than that one we teach in high school. I don't want to argue for or against either viewpoint, because at the end of the day they apply in different cases. You just have to learn to be flexible and use whichever one applies to the problem at hand. Just as the equation "2n = 3" doesn't have a solution in the integers, but has one in the rational numbers, the same series can diverge in one sense and converge in another.

Edit: I now see that topsquark made many of the same points above. Note that the function I mentioned above is none other than the Riemann zeta function topsquark mentions.
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October 30th, 2015, 04:44 PM   #4
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Thanks for taking the time to watch my video and responding.

The video that I am responding to claims there are many ways to get this -1/12 value, not just the Zeta function for -1.

This Wiki page describes the various methods: https://en.wikipedia.org/wiki/1_%2B_..._%2B_%E2%8B%AF

The obvious flaw on this Wiki page is that two methods, including the first and most prominent method, are types of smoothing where the graphs cross the y-axis at -1/8, not the -1/12 that the Wiki page claims. If you look at every other point on these graphs, such as x=0.5, x=1.5, x=2.5, x=3.5 and x=4.5, all match the graph for y=(x-0.5)(x+0.5)/2, and this has a y-intercept at -1/8.

Ramanujan came up with two methods to get this value, one was the informal method used by the Physicists in the video I was responding to.

One thing I would lke to know is which is best, the way the Physicists manipulate series or the way I do? I believe my way is more rigorous because it compares like terms and it cannot result in a fixed value being assigned to an endless series.

I understand that mathematicians 'define' a series to have a fixed value if it 'converges'. But to me this is a play on words, because the series cannot actually have a fixed value because it has no 'last term'. This would be fine if the terms were all zero, but they are not, and so the series cannot have a fixed value.

It appears that the Numberphile mathematicians don't really want to give a clear and pecise descriptrion of the relationship between the series 1+2+3+4+... and -1/12 because they are happy with the mystic association with 'infinity'.
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October 30th, 2015, 05:08 PM   #5
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You said all of the ways in which it converges shows it converging to the same value, -1/12.

But am I right in saying these methods only produce an answer of -1/12 because, through manipulations like 'subtraction', they are no longer considering just positive whole numbers, they have brought negative numbers into play (as I point out in the video)?

I particularly do not like these methods being used in physics. There are often many ways to calculate the same result, and if no 'tricks' were involved (i.e. nothing involving 'infinity') then what is actually going on in the real world might be understood with more clarity. Just because using 'infinity' happens to get the result you want, it does not validate the method used.

You did not lose me too much with 'analytic continuations'. I have read a little about them and I believe they are not necessarily unique. That is to say there may be a different continuation where the Zeta function at -1 is not -1/12.

I have made other points in my reply to +topsquark
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October 30th, 2015, 06:06 PM   #6
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Quote:
Originally Posted by Karma Peny View Post
I particularly do not like these methods being used in physics. There are often many ways to calculate the same result, and if no 'tricks' were involved (i.e. nothing involving 'infinity') then what is actually going on in the real world might be understood with more clarity. Just because using 'infinity' happens to get the result you want, it does not validate the method used.
The reason Physics has to use methods to find "values" for divergent sums is that the theories (or Mathematics applied to them) are not complete. In essence we need better theories. For now we simply have to use what works. As I said, it isn't comfortable, but it's what we have for now.

Quote:
Originally Posted by Karma Peny View Post
You did not lose me too much with 'analytic continuations'. I have read a little about them and I believe they are not necessarily unique. That is to say there may be a different continuation where the Zeta function at -1 is not -1/12.
The thing about analytic continuation is that if we analytically continue the Reimann-Zeta function into a region of the complex plane that contains s = -1 and it says that $\displaystyle \zeta (-1) = -1/12$ then all analytic continuations of the Reimann-Zeta function into any other area containing s = -1 will produce $\displaystyle \zeta (-1) = -1/12$. Analytic continuation is unique: ie. The representation of the function in different regions of the complex plane may be different but the values are the same for any regions in any overlap.

-Dan
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October 30th, 2015, 06:32 PM   #7
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Quote:
Originally Posted by topsquark View Post
The reason Physics has to use methods to find "values" for divergent sums is that the theories (or Mathematics applied to them) are not complete. In essence we need better theories. For now we simply have to use what works. As I said, it isn't comfortable, but it's what we have for now.
This is absolutely true. As a general rule mathematicians are not comfortable with the renormalizations that physicists use. But they do give sensible and practically useful answers to questions which seem to generate absurd (i.e., divergent) results otherwise.

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Analytic continuation is unique
I can't stress enough how important this is. Once you commit to one tiny piece of the function -- pick any complex number $z$ and any real $\varepsilon>0$ and all you need is the disc of radius $\varepsilon$ around $z$ -- you're committed to the whole thing in a unique way. This is unbelievable counterintuitive and the source of many of the powerful results in complex analysis. (I shouldn't have to point out how much of modern life relies directly or indirectly on complex analysis -- all electrical engineering, for example.)
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October 30th, 2015, 06:40 PM   #8
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Quote:
Originally Posted by Karma Peny View Post
I particularly do not like these methods being used in physics.
As a mathematician (rather than a physicist) I agree with you. But I can't deny the utility of the methods even if they don't yet have a firm justification for them. I expect that eventually there will be a rigorous treatment which will remove the need for this apparently ad-hoc method.

But don't be too confused by it: in practice, two physicists working on the same problem would both renormalize it the same way. It's not a case of getting anything you want!

Quote:
Originally Posted by Karma Peny View Post
You said all of the ways in which it converges shows it converging to the same value, -1/12.

But am I right in saying these methods only produce an answer of -1/12 because, through manipulations like 'subtraction', they are no longer considering just positive whole numbers, they have brought negative numbers into play (as I point out in the video)?
Not necessarily. As I said, there are many methods. Some expand into negative numbers, some even go into the complex numbers or into other weird kinds of mathematical entities, but there's a remarkable agreement about what the answers should be. For example, 'everyone' agrees that 1 + 4 + 9 + 16 + 25 + ... 'should' be equal to 0... and this is an important observation which has concrete ramifications for the (normal) prime numbers! Of course it's not obvious when the answer should be 0 and when it should be "this doesn't even make sense" but it seems that whenever it does make sense for it to be a number, the number that it should be is 0.
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October 30th, 2015, 07:06 PM   #9
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I see that there are some points I missed so I'll go back and address them.

Quote:
Originally Posted by Karma Peny View Post
This Wiki page describes the various methods: https://en.wikipedia.org/wiki/1_%2B_..._%2B_%E2%8B%AF

The obvious flaw on this Wiki page is that two methods, including the first and most prominent method, are types of smoothing where the graphs cross the y-axis at -1/8, not the -1/12 that the Wiki page claims. If you look at every other point on these graphs, such as x=0.5, x=1.5, x=2.5, x=3.5 and x=4.5, all match the graph for y=(x-0.5)(x+0.5)/2, and this has a y-intercept at -1/8.
I'm sure that's not the case. I'm not sure if the problem is a mistake on Wikipedia, in your own calculations, or in your assumptions. Perhaps if I have time I'll look into it.

Quote:
Originally Posted by Karma Peny View Post
Ramanujan came up with two methods to get this value, one was the informal method used by the Physicists in the video I was responding to.
I don't know anything about that. The method I'm talking about, Ramanujan summation, is entirely rigorous. I didn't even know resummation was known when Ramanujan was alive. (It may very well have been, I know little about the history of science.)

Quote:
Originally Posted by Karma Peny View Post
One thing I would lke to know is which is best, the way the Physicists manipulate series or the way I do? I believe my way is more rigorous because it compares like terms and it cannot result in a fixed value being assigned to an endless series.
I don't think there's a best method, just ones which are appropriate or inappropriate for the task at hand. In the ordinary Euclidean sense the geometric series 1/2 + 1/4 + 1/8 + ... converges and has the value 1, but in 2-adic analysis it diverges.

The methods used to compute 1 + 2 + 3 + ... = -1/12 are completely rigorous, but they may not be appropriate for a given task.

Quote:
Originally Posted by Karma Peny View Post
I understand that mathematicians 'define' a series to have a fixed value if it 'converges'. But to me this is a play on words, because the series cannot actually have a fixed value because it has no 'last term'.
Well, mathematicians have a very precise definition of what they mean when they say that. If it's not appropriate for what you're doing, then don't use it. But infinite sums of nonzero terms have a tremendous number of applications, in finance, charity, engineering, science, mathematics, etc.

Quote:
Originally Posted by Karma Peny View Post
It appears that the Numberphile mathematicians don't really want to give a clear and pecise descriptrion of the relationship between the series 1+2+3+4+... and -1/12 because they are happy with the mystic association with 'infinity'.
Numberphile, while very enjoyable, is targeted toward a popular audience and it's hard to be rigorous at that level. They try, but when they have a choice between being more rigorous and more approachable they usually take the latter.
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October 30th, 2015, 08:01 PM   #10
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I have two comments to make. The first follows on from topsquark's comments on analytic continuation, the second is my recently acquire view on assigning values to divergent series.

Analytic continuations are unique. We have an analytic function that is defined in a region. Actually, the way to think of it is that we have a representation of an analytic function, and that representation is valid only within a particular region. The continuation is effected by finding another way of representing that same function within a part of that region. If the two representations coincide exactly, then they are necessarily the same function. But this second representation may provide a legitimate value over a different (but overlapping) region. In this case we can use the value of the second representation over all of the region for which it is valid. The continuity of the region over which each representation is defined guarantees that the different representations are exactly equivalent and their validity and agreement over a shared region mean that the function described over the union of the regions is also analytic.

I have realised that we think of our method of assigning a value to convergent series in the wrong way. We tend to think that the value of such a series is correct because the partial sums get closer and closer to that value. My contention is that we should not forget that the partial sums never reach that value.

Cantor extracted the salient features of counting (finite collections) and then applied them to a different domain (infinite collections). In doing so he found results about countability and cardinality of infinite sets.

Similarly, the techniques of generic summation (and others) extract the salient features of the summation of convergent series and apply them to non-convergent series. As a result the two processes are equivalent.

The fact that the value we assign to convergent sums is close to the value we see the partial sums nearing is good, but not essential. What matters is that we have followed the rules without making any assumptions about the nature of infinite sums. Similarly, when we count finite collections. When we then count infinite sets, we can be sure that we are getting sound results even though some of them are counter-intuitive. This is precisely because we do not let our intuitions override the results. Our intuition related to infinite operations is necessarily unreliable because we have no direct experience of such operations. Thus, when we stick to the rules of assigning values to infinite sums, we can be sure that we attain good answers even if they are counter-intuitive.

I am not necessarily comfortable with the idea that the value we assign to a sum actually represents the result of that sum. It is easily argued that the concept of a result of an infinite sum makes no sense anyway since you can never complete the sum. I simply think of it as assigning a value to a mathematical object. The fact that it looks like a sum is neither here nor there: it's just an object with a value.
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