# The relationship between 1+2+3+4+... and -1/12

#### skipjack

Forum Staff
The obvious flaw on this Wiki page . . . 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.
You seem to be misreading the graph. The parabola shown is for y = xÂ²/2 - 1/12.

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#### v8archie

Math Team
I've just watched your video, which while interesting does contain one glaring problem. You have arbitrarily decided that your partial sums should be symmetrical and, even worse, that these finite "negative" partial sums should influence the positive infinite sum.

Admittedly, the methods of assigning values to these sums also make assumptions, but they are minimal assumptions designed to mirror the approach we take with convergent sums.

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#### Karma Peny

You have arbitrarily decided that your partial sums should be symmetrical
By the way, many thanks for watching my video.

I had a good reason to choose symmetry.

If you look at the graph for x(x+1)/2 but just consider positive values of x, and then you 'remove the diverging part', then nothing remains.
This matches what you would expect from the partial sum expression. The fixed part of the partial sum expression is what is called the 'limit' for a 'converging' series. It is my understanding that these methods are trying to get this value for a diverging series, and I guess it might not always be so easy to find the fixed part of the partial sum expression.

The reason I wanted symmetry was to ensure that methods that 'remove the diverging part' no longer get this value of -1/12, which they would not get if they were just considering the positive values of x.

Symmetry allows these other methods to arrive at the the result of zero, instead of getting this -1/12 value resulting from the skew. Hence they no longer contradict the value obtained from the partial sum expression.

This is the same reason I wanted symmetry of the Zeta function, like this would produce:

$$\sum_{n=1,2,3...}\frac{s}{|s|.n|^s|}$$

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#### v8archie

Math Team
It is my understanding that these methods are trying to get this value for a diverging series
No, absolutely not. As far as I understand, these methods are intended only to assign a value to an infinite series - and it doesn't matter whether it is a convergent one or a divergent one. Even more importantly, they are intended to assign that value based on very limited assumptions. Series such as $1-1+1-1+1-1+\ldots$ are evaluated using only arithmetic properties that have not been shown to be invalid in convergent infinite sums. Namely, that
• if you take the first term of the series and add on the value assigned to the rest of the infinite series, you should get the value assigned to the whole of the infinite series $$a_0 + S(a_1+a_2+a_3+\ldots) = S(a_0+a_1+a_2+a_3+\ldots)$$
• that the way in which you assign a value to an infinite series should exhibit linearity. $$S\big((xa_0 + yb_0) + (xa_1 + yb_1) + \ldots\big) = xS(a_0 + a_1 + \ldots) + yS(b_0 + b_1 + \ldots)$$
This approach doesn't work for some series (that are used in deriving $1+2+3+\ldots = -\frac1{12}$) so analytic continuation is used instead. I think we've already covered the validity of that idea.
The reason I wanted symmetry was to ensure that methods that 'remove the diverging part' no longer get this value of -1/12, which they would not get if they were just considering the positive values of x.
So not only do you wish to arbitrarily impose symmetry, you also wish to arbitrarily decide the value that should be assigned to a sum. The methods we are discussing here were not designed to validate a particular value, but rather to see if there was/is a value that has any claim to be reasonable based only on the evidence obtained by following the simple rules.

Symmetry allows these other methods to arrive at the the result of zero, instead of getting this -1/12 value resulting from the skew. Hence they no longer contradict the value obtained from the partial sum expression.
The claim you make here is that infinite sums behave like finite sums. But they don't! The alternating harmonic series (or any other conditionally convergent series) can be made to have whatever value you like if you rearrange the terms. This doesn't happen with finite sums, so we know that the two are different. The techniques we are talking about, accept that there are differences and refuse completely to allow us to impose our preconceptions that are based on finite arithmetic onto infinite sums.

These techniques obey rules developed from the rules of arithmetic and analysis by discarding the rules of the finite realm that demonstrably no longer apply in the infinite realm. If you wish to attack the results, you'll have to find flaws in the formulation of the rules, not in the results that spring from them.

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#### Karma Peny

and it doesn't matter whether it is a convergent one or a divergent one.
Agreed. So if we talk about 'the fixed part of the partial sum expression' then this value can be calculated whatever type of series it is. It just so happens that this value will equal the limit for 'converging' series. For a 'diverging' series we could say this value is 'what the limit would be if the series converged'.

Even more importantly, they are intended to assign that value based on very limited assumptions. Series such as $1-1+1-1+1-1+\ldots$ are evaluated using only arithmetic properties that have not been shown to be invalid in convergent infinite sums. Namely, that
• if you take the first term of the series and add on the value assigned to the rest of the infinite series, you should get the value assigned to the whole of the infinite series

• I have two problems with this. Firstly, it is not at all clear what the meaning of this value will be, as it just seems to be the result of some very non-rigorous manipulations. Secondly, there is the assumption that 'the rest of' an 'infinite series' can be assigned a fixed value. This is the reason I decided to do my own video in which I manipulate series with respect to the n-th term.

So not only do you wish to arbitrarily impose symmetry, you also wish to arbitrarily decide the value that should be assigned to a sum.
No, I am being very clear that the value I am talking about is the fixed part of the partial sum expression. Can you be this clear about whatever value you are talking about?

The methods we are discussing here were not designed to validate a particular value, but rather to see if there was/is a value that has any claim to be reasonable based only on the evidence obtained by following the simple rules

What does 'a value that has any claim to be reasonable' mean?

The claim you make here is that infinite sums behave like finite sums
No I don't. I claim that if a series is endless it should treated as being endless, not as if it were a fixed value. Finite series have fixed values, endless series don't. They should be treated differently.

#### v8archie

Math Team
Agreed. So if we talk about 'the fixed part of the partial sum expression' then this value can be calculated whatever type of series it is. It just so happens that this value will equal the limit for 'converging' series. For a 'diverging' series we could say this value is 'what the limit would be if the series converged'.
The problem with this idea is that the partial sums are by definition finite. The idea behind the summation of these series is that infinite sums are different animals to finite ones. I am, of course, aware that for convergent series we use the one to evaluate the other, but this is a particular class of infinite sum. It doesn't mean that we either can or should use the same technique for all infinite sums.

You are also fixed on the concept of evaluating the sum. This is expressly what the ideas I am talking about do not do. They assign a value to a series, it is not necessary to believe that it has any link with evaluating the series.
I have two problems with this. Firstly, it is not at all clear what the meaning of this value will be
Why does there need to be a meaning? Perhaps there is one that we don't know - after all, physics seems to work along these lines in some sense. One possible explanation is that these sums are taking place on the Riemann Sphere and follow the sphere round past infinity. This is young maths - all the answers are not there yet. As to the rigour, this is a formally worked theory. As I said before, you may challenge the assumptions, but the results are watertight applications of those rules.
Secondly, there is the assumption that 'the rest of' an 'infinite series' can be assigned a fixed value.
There is no assumption that the rest of the series can be assigned a value. Only that any series can be assigned a value. And it is fine for that value to be "infinity" - as it is for the sum $1+1+1+\ldots$.

In the statement of the rule that I gave, there are two (related) series $a_0+a_1+a_2+\ldots$ and $a_1+a_2+\ldots$. The "rest of the series" phrase was intended to highlight the relation between the two, no more.

The basic assumption here is less mathematical than lexical. Given a mathematical object of a certain structure, we assert that we can give it a value in the extended real numbers. This value need not have any relation to evaluating the sum that the object looks like, we only insist that certain rules are followed in generating the value that we assign to the object.

No, I am being very clear that the value I am talking about is the fixed part of the partial sum expression.
Again, you are insisting that infinite sums must have a particular relation to finite sums in all cases. But we know that such sums are different animals. In the case of convergent infinite sums, we know that there is a close relation which exists by virtue of the terms of the sums becoming arbitrarily small. But that relation doesn't exist without that condition.

What does 'a value that has any claim to be reasonable' mean?
Well, it's a deliberately vague phrase. But some features that make the claim reasonable are that the value is common to many different approaches to the problem of assigning a value to the object and that the value is generated from a clear and reasonable set of assumptions, by following clearly defined and reasonable rules.

No I don't. I claim that if a series is endless it should treated as being endless, not as if it were a fixed value. Finite series have fixed values, endless series don't. They should be treated differently.
And yet you claim that the finite series should dictate the value of the infinite series. Again, we do not need to consider this as a process of evaluation. We might even come to agreement on the idea that an infinite sum cannot be evaluated, precisely because it is endless. But such a statement does not stop us from assigning a value to an object. We do this all the time: we assign a value to the limit of a function or an expression. That limit is not an evaluation of the function or the expression, it is a mathematical object.

This whole subject can be seen as an extreme finitist approach to infinite sums - we explicitly abandon any attempt at evaluating an infinite sum inf favour of finding a finite value by which the sum can be represented in finite arithmetic. This is in direct contrast to our usual way of viewing convergent sums where we are in the habit of claiming that the limit is the evaluation of the infinite series.

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#### v8archie

Math Team
I should also add that a "reasonable" value for a convergent series is it's limit. This idea was a reason for the vagueness of that sentence.

It occurs to me that we might compare this theory with calculus under the hyperreal numbers. Both are formalisations of non-rigours methods used in the past. I'd claim that this one is superior to the hyperrals, because of the way definitions are stated. The calculus incorporates an error correction into the definitions.

#### Karma Peny

The problem with this idea is that the partial sums are by definition finite. The idea behind the summation of these series is that infinite sums are different animals to finite ones..
Just because a series is endless, it does not mean it is 'infinite'. Once you assume or 'define' that it exists in its entirety you have problems. The first problem is you cannot explain how it can exist in its entirety and yet have no 'last term'. If this is not a contradiction then it is at least a 'paradox'. Another problem you have is that you can prove the value does not end, since for any given term there will be term after it that will have a non-zero value. To me, this appears to prove that the object cannot exist in 'its entirety'.

Why not avoid these problems and simply treat an endless object as an endless object (like I do), rather than assume 'infinity' must exist?

You are also fixed on the concept of evaluating the sum.
I can't believe you are saying this. I keep stressing over and over that an endless series (where the endless terms are non-zero) cannot have a sum. If anything, I am fixed on NOT evaluating the sum

They assign a value to a series, it is not necessary to believe that it has any link with evaluating the series.
Yes, but these methods claim to 'remove the infinite parts' and so we are left with the notition that the resulting value is something mysterious that is related to 'infinity', as a maths professor points out in this video:

Why -1/12 is a gold nugget: https://www.youtube.com/watch?v=0Oazb7IWzbA

In this video he says about the sum of the powers of natural numbers:
"Why did I skip squares... the answer is more surprising... for all even values you actually get zeroes, and for odd values you get some rational number"

But to me this is not surprising at all. If you consider the graph of the partial sum expression, then for even values you will get a reflection about the y-axis but for odd powers you will not.

He also says:
"We are not content with just saying that there is some magic over there, there is magic, but we always want to explain it"

And:
"does it mean that in some sense it is, there is a context in which this sum, this infinite sum, is mysteriously minus one twelfth? I'm not sure."

It appears to me that mainstream mathematics regard this -1/12 value as being shrouded in the mystery and magic that comes from using methods that claim to work with 'infinity' in some way.

Why does there need to be a meaning?
Because if you then proceeed to use it, you will not understand what you are doing. And if you don't proceed to use it, then it is of no value.

There is no assumption that the rest of the series can be assigned a value. Only that any series can be assigned a value. And it is fine for that value to be "infinity" - as it is for the sum $1+1+1+\ldots$
But what method have you used to get this 'value' of 'infinity'? Since the partial sum here is simply n, the fixed part should be zero, not 'infinity' whatever that means. Also, this partial sum expression will result in symmetry about the y-axis and it will intersect at y=0, so I think the summation methods will agree with my answer rather than yours.

We might even come to agreement on the idea that an infinite sum cannot be evaluated, precisely because it is endless. But such a statement does not stop us from assigning a value to an object.
Yes, I do hope we can agree; that would be nice. I have no objection to attaching a value to an object, just as long as we are absolutely clear about what it means.

This is in direct contrast to our usual way of viewing convergent sums where we are in the habit of claiming that the limit is the evaluation of the infinite series.
Being a habit does not make it correct.

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#### v8archie

Math Team
Just because a series is endless, it does not mean it is 'infinite'.
Erm... yes it does. That is the meaning of infinite. I'll attempt to understand your distinction for the benefit of the discussion though. I imagine that you consider these objects to be finite sums that don't have an endpoint. This explains your fixation with partial sums, as to you it appears that these sums are partial sums. I'm not sure where we can go with that, because it sounds bonkers to me.
... To me, this appears to prove that the object cannot exist in 'its entirety'.
Well a line is infinite (or at least, it is endless), but we have no trouble representing the entire line as a set $\{\vec a + t \vec b\}$ or an equation $(y-y_0)=m(x-x_0)$, and we quite happily consider limits to be an object despite the fact that their construction explicitly ensures that they have no endpoint. Infinite "sums" need be no different, especially if you can stop thinking of them as sequences of addition operations and instead think of them as objects.
Why not avoid these problems and simply treat an endless object as an endless object (like I do), rather than assume 'infinity' must exist?
Actually, I am not assuming that "infinity" exists (other than as a token that is assigned to infinite sums that cannot be assigned a finite value). I am simply considering them to be an object. I call them infinite (or endless) objects as a way to differentiate them from the finite sums that are subject to different rules such as associativity and commutativity.
I keep stressing over and over that an endless series (where the endless terms are non-zero) cannot have a sum. If anything, I am fixed on NOT evaluating the sum
And yet you persist in evaluating partial sums and claiming that they have a direct relation to the infinite object.
these methods claim to 'remove the infinite parts'
I don't agree with this. I think that it's Numberphile's way of simplifying the idea for a more general audience. Either that, or the physicists interviewed don't understand the mathematical basis of the ideas. The idea of "removing the infinite parts" suggests that you have been adding the elements of the sum. My main reference on this is Carl Bender's lectures, where he does say that adding up the terms of infinite series is stupid.
If you consider the graph of the partial sum expression, then for even values you will get a reflection about the y-axis but for odd powers you will not.
I'm actually quite interested in your graph - although not for the reasons you are. Like you, I suspect that it might give some insight into why the values are as they are, but I am curious as to whether that might lead to some realistic understanding of the physics in which this stuff is used.

"We are not content with just saying that there is some magic over there, there is magic, but we always want to explain it"... "does it mean that in some sense it is, there is a context in which this sum, this infinite sum, is mysteriously minus one twelfth? I'm not sure."
The first quote sounds like it is a reference to Clark's third law. The second is essentially what we are talking about. There are obviously very good reasons to suppose that the sum of a bunch of positive integers can't be a negative non-integer, but the Reimann sphere suggests possibilities for why it might. My position at the moment is that we are not adding an infinite number of things together, so any rules of closure don't apply. But then, we already know that the limit of a sequence of rationals can be irrational, so why should such rules apply?

Because if you then proceed to use it, you will not understand what you are doing. And if you don't proceed to use it, then it is of no value.
Well, physicists openly admit that they don't know why it works, but they also point out that the predictions made using this stuff are good. They hope to understand it later. I'd agree with that - this stuff is close to frontier of mathematical understanding to, so there's every reason to expect that it should be incompletely understood. But I'd also say that, in a purely mathematical context it really doesn't matter. Abstract Algebra is utterly devoid of meaning to me, but that doesn't stop me from using numbers, nor does it make mathematics devoid of value. The undefined terms of any system (such as the lines and points in geometry) explicitly do not have any intrinsic meaning. We impose a meaning when we identify a relationship between them and something that we wish to study. And this relationship then means that all theorems involving those objects ought to have a meaning as a result, but it is far from obvious in all cases what that meaning might be.
But what method have you used to get this 'value' of 'infinity'?
Well the derivations goes:
We are going to assign a value on the Riemann Sphere $s$ to $1+1+1+\ldots$ using generic summation. Thus \begin{aligned}s &= S(1+1+1+...) \\ s &= 1+ S(1+1+1+...) &\text{by the first rule I gave above} \\ s &= 1 + s \\ \implies s = \infty\end{aligned}​
Now, there's a lot of "fudge" in that. But the point is not in how we get to that answer, but the fact that we accept the answer that we get. Essentially it means that "this technique is not powerful enough to assign a finite value to this series". It turns out that analytic continuation is powerful enough to do that, and gives us the value $-{1 \over 2}$.

In essence, I believe that you need to stop concentrating on finite partial sums. They work well as a tool for convergent series, but are not powerful enough to assign a finite value to divergent series. I also believe that you need to accept mathematics for what it is, rather than for how it can be interpreted.

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#### Karma Peny

Erm... yes it does. That is the meaning of infinite. I'll attempt to understand your distinction for the benefit of the discussion though.
Then you say

it sounds bonkers to me.
Nice attempt at understanding the distinction. You completely ignored the 'paradoxes' I mentioned and went off to talk about sets.

And yet you persist in evaluating partial sums and claiming that they have a direct relation to the infinite object.
Well no, I don't believe in your 'infinite object', but of course the partial sum expressions ARE VERY relevant to where values like -1/12 come from. There is no such thing as an 'infinite' object that has 'infinitely many' elements. (NOTE: I would have no issue with 'infinite' as meaning 'endless' if people did not assume that it means 'infinitely many' is a valid concept, but they do, they ALWAYS do).

After watching that Numberphile video (link in my previous post), I decided to derrive the partial sums for

$$1^2 + 2^2 + 3^2 + 4^2 +...$$

and for

$$1^3 + 2^3 + 3^3 + 4^3 + ...$$

then I plotted them on a graph.

I also took the definite integrals of these expressions between 0 and -1, which is where the relevant part of the 'skew' occurs. Guess what, I got zero and plus 1/120 just like in the video.

So, either it is one huge coincidence that these figures match or the Extreme Finitism approach has correctly identified where these â€˜mysteriousâ€™ values are coming from (& it surely isnâ€™t infinity!).

You can keep your mysteries and your belief in 'infinity' - you must do whatever makes you happy, but I prefer to understand what I'm doing. It has been enjoyable discussing this with you, I hope I don't sound too disgruntled. We all must live with our own beliefs and not chastise others for their beliefs.

Many thanks once again, and goodbye.

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