# The Sum of All the Natural Numbers

by AlfieD
Tags: bosonic, infinite series, natural, numbers, string theory, zeta function
PF Gold
P: 735
 Quote by 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
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."
Mentor
P: 14,243
 Quote by jbunniii 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/2010/0...-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.
 Sci Advisor P: 7,909 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.
Sci Advisor
P: 7,909
 Quote by D H That said, the youtube video cited in the original post is an example of "physicists doing math (badly)".
I just came across the Cesaro sum http://en.wikipedia.org/wiki/Summati...ndi%27s_series which Wikipedia says is "rigourous", and seems very close to what was being presented in the OP's video.
 Mentor P: 14,243 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.
 Mentor P: 4,499 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.
 PF Gold P: 1,967 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
Sci Advisor
P: 7,909
 Quote by Office_Shredder 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?
P: 764
 Quote by atyy Is it right to say that 1+2+3+... is also not Abel summable?
Correct. It is mentioned in Wikipedia page as well.
 P: 1,180 '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.
P: 68
 Quote by 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"'
Hmmm, interesting!
 P: 1 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?
 P: 109 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.

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