# The Sum of All the Natural Numbers

#### mfb

Mentor
@Office_Shredder and FlexGunship: See post 14, where I got a contradiction with the same methods used in the video.

#### FlexGunship

Gold Member
FYI, I've been away for the last couple of days. I'm still looking at this. I'min way over my head, but it still makes for a fun intellectual challenge.

My latest notebook scribbling seem to imply fundamentally different behaviors for infinite sums that have alternating signs and sums that don't. This leads me to think that sums with non-alternating signs have some complex component to them that is not being written explicitly. This complex component becomes obvious when you "shift by 1 term". Certainly not a valid argument (you can always say, "fine, that works with complex terms, but now do the sum without them"), but it sure makes for some good brain exercise.

However, if you accept that a "shift by 1" doesn't work, and try a "shift by 2"...

$S = +1+2+3+4+5+...$
$S = -0-0-1-2-3-4-5-...$
--------------------------------------
$0 = +1+2+2+2+2+2+...$

$0 = 1 + 2(T)$
$-\dfrac{1}{2}=T$

This result might also be arbitrary (or arbitrarily wrong), but at least it matches:

$S_1 = 1-1+1-1+1-1+... = \dfrac{1}{2}$ via the following:

$+(T = 1+1+1+1+1+1+... = -1/2)$
$-(S_1 = 1-1+1-1+1-1+... = 1/2)$
---------------------------------------
$T-S_1 = 0+2+0+2+0+2+0+2-... = -1$
$T-S_1 = 2(1+1+1+1+1+1+1+...) = -1$
$T-S_1 = 2T=-1$
So, again: $T=-\dfrac{1}{2}$

The fundamental problem is that, in both cases, you have to "pretend" there are intervening complex terms in non-alternating infinite sums and just arbitrarily avoid mixing them.

And even more problematically... I have no idea what the "generally accepted" sum of 1+1+1+1+1+... is because I don't know how to enter it into Wolfram-Alpha and there's no Wikipedia page on it... and -1/2 doesn't make an iota of sense.

#### atyy

For those interested in the abuse of mathematics, here's David Tong's string theory notes. The relevant pages to read are his p39-40 and p85. http://www.damtp.cam.ac.uk/user/tong/string.html

#### marcusl

Gold Member
E. T. Jaynes addresses exactly this sort of mis-summation of infinite series in his book Probability Theory: The Logic of Science. Section 15.2 opens this way:

As a kind of introduction to fallacious reasoning with infinite sets, we recall an old parlor game by which you can prove that any given infinite series $S=\sum_i a_i$ converges to any number x that your victim chooses.

So we aren't limited to -1/12; he goes on to show how to sum it to anything you want. He identifies the crux of the fallacy thus:

Apply the ordinary processes of arithmetic and analysis only to expressions with a finite number n of terms. Then after the calculation is done, observe how the resulting finite expressions behave as n increases indefinitely. Put more succinctly, passage to a limit should always be the last operation, not the first. In case of doubt, this is the only safe way to proceed.

The jokers in the video violate this by manipulating and "evaluating" infinite sums without considering the proper limiting process.

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#### FlexGunship

Gold Member
For those interested in the abuse of mathematics, here's David Tong's string theory notes. The relevant pages to read are his p39-40 and p85. http://www.damtp.cam.ac.uk/user/tong/string.html
So, this strikes me as representative of the discussion thus far.

I read p39-40, and I'm okay with it. I don't know zeta function regularization, but I accept that everyone is getting -1/12 out of this particular operation, and that everyone acknowledges that this is famously unconvincing. Fine.

However, the implication is that anyone working in the field of string theory must first dispose of common mathematical practice prior to "getting anything done" yet some of the giants of mathematics (like Euler, Riemann, and Ramanujan) seem to concur on this particular point.

But there's a genuine irony here. Physics is applied mathematics. Specifically, it is the application of mathematics to our observation of reality. Often, physics calls upon mathematics to formalize and/or generalize an observation. But in this ONE area, the discussion proceeds like this:

Physics: "Hey, math!"
Math: "What?"
Physics: "I've got this thing. It looks like infinity but, uh, it needs to be -1/12. I know that's crazy and random... but..."
Math: "Yeah, I have one of those."
Physics: "REALLY?!"
Math: "Yeah."
Physics: "Well, let me use it!"
Math: "Naw, it's just a trick I know."
Physics: "Oh..."
Math: "Well, actually, it's like... three tricks I know... super important tricks!"
Physics: "Oh?!"
Math: "But you still can't use it."

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#### D H

Staff Emeritus
I don't see anyone advocating this.
Euler, Hardy, Ramanujan advocated exactly this. More recently, here's Terrance Tao advocating this: http://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/

That said, the youtube video cited in the original post is an example of "physicists doing math (badly)". To see "mathematicians doing math (formally)", see the link that I posted. Or see the other links where $1+2+3+4+\ldots = -1/12$ is formally defined via zeta function regularization. That's an example of "physicists doing math (correctly, surprisingly)".

What you cannot do is manipulate those formal sums on a term-by-term basis. Do that and it's easy to arrive at contradictions. Unfortunately, that's exactly what was done in video cited in the OP, hence my label "physicists doing math (badly)." Just because their manipulations happened to arrive at the formal result does not mean that what they did was valid. It isn't.

#### atyy

So we have $1 + 2 + 3 + ... = -1/12$ being made sensible by the Riemann zeta function $\zeta(-1)=-1/12$.

How about the starting point in the video $1 - 1 + 1 -1 + ... = 1/2$ ? After some googling, I found the the Dirichlet eta function $\eta(0) = 1/2$. Wikipedia calls this the Abel sum of Grandi's series. Is that the right notion here?

Edit: Looking at D H's link in his post #56 to Terry Tao's article, it looks like it is.

#### D H

Staff Emeritus
Nope. Everything after the 2:50 mark in that video is invalid. You can't do that with conditionally convergent series, let alone divergent ones.

#### Office_Shredder

Staff Emeritus
Gold Member
atyy, Cesaro summation is a rigorous method of assigning values to series. The series 1+2+3+.... is not Cesaro summable though, so it doesn't help with the original problem. It is used in some areas to great effect; for example partial sums of Fourier series can be quite bad on continuous functions even (diverging everywhere), but the Cesaro sum ends up converging for functions that are way worse than continuous. It ends up that when you want to prove results about Fourier transforms it's a lot better to consider the Cesaro sum of the Fourier series rather than the regular sum.

Notice how the wikipedia article then goes on in a later section to describe a whole host of issues in which inserting zeroes into sums can change the Cesaro sum, so when doing Cesaro summation you have to be extremely careful that you are being actually rigorous, and not just taking the word rigorous and slapping it onto a bad argument.

Look at how he adds S2 to S2. He writes one series of numbers (from 1 to 7 in the example given)and writes an identical second series of numbers which he places below the first series.He then pushes one of the two series along by one and concludes that the sum is given by 1-1+1-1 etc .The sum, however, is not given by that. In the example given where the last digit equals 7 the sum is equal to 8. He ignored the seven at the end of the pushed along series. This number 7 was not overlapping with any other number.
When the non overlapping number at the end of the series is taken into account one can see that 2s2 approaches infinity as the number of numbers in each series approaches infinity (plus or minus infinity depending on the number of digits). 2s2 has numerical values which increase with the length of the series as follows:
+2,-2,+4,-4,+6,-6, etc

#### atyy

atyy, Cesaro summation is a rigorous method of assigning values to series. The series 1+2+3+.... is not Cesaro summable though, so it doesn't help with the original problem. It is used in some areas to great effect; for example partial sums of Fourier series can be quite bad on continuous functions even (diverging everywhere), but the Cesaro sum ends up converging for functions that are way worse than continuous. It ends up that when you want to prove results about Fourier transforms it's a lot better to consider the Cesaro sum of the Fourier series rather than the regular sum.

Notice how the wikipedia article then goes on in a later section to describe a whole host of issues in which inserting zeroes into sums can change the Cesaro sum, so when doing Cesaro summation you have to be extremely careful that you are being actually rigorous, and not just taking the word rigorous and slapping it onto a bad argument.
Is it right to say that 1+2+3+... is also not Abel summable?

So one really has to use zeta regularization, or the smooth cut-off mentioned in the link that D H gave in #56?

#### 1MileCrash

'After taking a course in mathematical physics, I wanted to know the real difference between Mathematicians and Physicists. A professor friend told me "A Physicist is someone who averages the first 3 terms of a divergent series"'

I thought you'd all enjoy that one.

#### AlfieD

'After taking a course in mathematical physics, I wanted to know the real difference between Mathematicians and Physicists. A professor friend told me "A Physicist is someone who averages the first 3 terms of a divergent series"'
Hmmm, interesting!

#### pjantoniopj

Here's where i saw the first mistake with Numberphile;

Given:
S1 = 1 - 1 + 1 - 1 + 1 - 1 + ...

Numberphile folks concluded that they could not just do the regular math (addition, subtraction) to obtain S1 = 0.
So they introduced their next step; add S1 to S1(shifted to the right)

S1 = 1 - 1 + 1 - 1 + 1 - 1 + ...
S1 = 1 - 1 + 1 - 1 + 1 - 1 + ...

And then, magically, they concluded that Yes, now we can to regular math here. They proceeded to add and subtract in a vertical direction.
Wait, what? You couldn't add and subtract horizontally but you can add and subtract vertically?

If you can add 1+0, then -1+1, then 1-1, then -1+1, etc., FOREVER, then why couldn't you do that in the first place with just S1?

#### cryptist

Algebraic manipulations on the video may be wrong, however, it is TRUE that finite and physically useful results can be obtained from divergent sums. In fact, mathematicians themselves are using it. There is a technique called analytic continuation that provides an extention of the domains of analytic functions. Note that, these odd results also consistent within the complex analysis. These divergent sums do not have contradictory finite results, they are actually valid!

Secondly, Casimir effect is a physical phenomenon that experimentally verified and well understood. In these effect, infinite summation of frequencies gives Zeta[-3]=1/120 where Zeta[-3] is a divergent sum and this result is verified experimentally. So in nature infinite summation of some "things" can REALLY give a finite result, which is consistent with mathematics.