Dear Stefano,

    This is a very interesting result of numerical calculus that Ramanujan sent to Hardy.

     R=1+2+3+4+5+6+….

      May be written as

        4R= 4+ 8+ …

      And thus we have

      -3R= 1-2+3-4+…. =1/(1+1)2

     Which implies that

      R=-1/12

   as you have written. This is a quite well result that in fact solves the Abel sum

                     Σ (-1)r r

   Dear Stefano,

  I have written the strange result that you present in a simple and clear form where was its origin. Another thing is that I believe it and even I can understand it. It seems quite unreasonable think the natural number addition can lead to a negative number as its outcome. But seeing my previous calculations you can observe that there are two nonusual manipulations:

  1. The operation

                   ∞-4∞=-3∞

is not following straightforward what we understand by infinite: for example, 4 infinite is infinite. Thus the previous operation could giver zero, instead of -3 infinite.

  2. Grouping the different terms of a serie is not so simple to justify too.

Dear Daniel,

I know but replacing a positive integer infinite sum with a negative finite fractionary number which is just the opposite of one could imagine is not  acceptable as a result. Don't you think this process hides something?

Think about the cauchy integrals and the laplace transforms they are based on complex numbers. For example The L tranform of the dirac delta is "1"....

The Riemann Zeta Function in its analytic continuation (because otherwise it would not be defined in the half plane of the negatives), applies a sort of a trasformation, but -1/12 is not the actual result of the series, but only the transform of the series by the means of the RZF because the series is divergent.

Dear Remi,

"This question rings a bell. I can't remember if it was asked a few years ago on RG or elsewhere. I shall look at my past questions to see if I can find it."

definitely yes, but here I would like to know what people think can be behind that.

     Sorry. I have forgotten to add 1 in the exponent

                Σ (-1)r+1 r

        Dear Stefano,

The problem is that there are no convergent series that you manipulate to obtain a finite value, but this is not unique. For instance, instead of substract directly 4R you can do it in two steps

         R−2R=1+3+5+

give you only the odd numbers and now if you rest 2R again, you have two possibilities depending of how you do it

        (R-2R)-2R=-1-1-1-1.... ( starting 1-2)

        (R-2R)-2R=1+1+1+1....=ζ(0)=-1/2(starting 3-2)

In this case we have R=1/6 .

    Dear Remi,

    Please, could you explain what is false? Your expression adding parenthesis or the mine? What is false?

Daniel,

it simply depends where you put the parenthesis...

The associative property for finite sums does not hold for infinite series, because  different results are obtanied...

https://www.youtube.com/watch?v=0Oazb7IWzbA

   Dear Stefano,

   there are more things than the associative property. Notice that we are selecting subseries very specials for obtaining an  finite value of the convergence. In the previous case we have also the Abel's sum to be introduced.

I have not looked into Euler's method, but I'm deeply skeptical of any particular value.  Renormalization schemes are really arbitrary.  They don't necessarily have a deep reason.  It is a matter of the right answer divided by the answer I got times the answer I got equals the right answer (what we used to call a fudge factor).  If one looks at all possible ways to generate arbitrary factors, there is a good chance one of them will equal the fudge factor by coincidence.

Using a variation of Gauss' method http://mathcentral.uregina.ca/qq/database/qq.02.06/jo1.html , I can get a variety of different answers for the sum of natural numbers.  For example, write the natural numbers as a series according to some algorithm such as

{N1} = {... -4 -2/3 -1/3 -3 -1/2 -2 -1 0 1 2 1/2 3 1/3 2/3 4 ...}

Now reverse that series

{N1r} = {... 4 2/3 1/3 3 1/2 2 1 0 - 1 -2 -1/2 -3 -1/3 -2/3 -4 ...}

It appears that SUM{N1} = 1/2 SUM{N1}+SUM{N1r} = 0

It also appears that one might get other sums by shifting N1r to the left or right, though the way I've written it (algorithmically easy to show as complete) does not lend itself to calculating these sums.  They might not be determinate.

If the sum is truly not determinate, and method dependent, then Euler's method might be provably correct without proving other sums are incorrect.  This is closely related to the Godel theorem, though not exactly the same.  It is a proposition that might have multiple correct answers. 

You can see this more easily by just looking at the sum of the integers.  If the reverse sequence is aligned at zero, the sum is zero.  If mis-aligned by one place it is naively + or - half of the number of whole numbers (i.e. infinity-1 I'll call it), and if mis-aligned by two places it is twice that, etc. 

   Dear Robert,

   Your list is not of natural numbers, you in fact integers Z and fractional Q numbers. Thus it is normal that the order is important in your case. What is much more strange is that the addition of natural numbers is not a closed operation and it falls in the negative fractional number -1/12.

    These surprising results belongs to the divergent series where Ramanujan and Hardy were speciallists. In physics -1/12 can be taken as a cutoff for distinguishing a within an effective theory. In fact, that allow to distinguish different scales. Notice that in topological theories there is not possible to obtain always a cutoff. Thus the regularization allows to remove different infinities of a non convergent series.

    The renormalization, for example in QED, goes a step further as introducing self-interaction concepts to define new operators.

    In any case, it seems quite arbitrary to work with non convergent series but in fact there are many fixed rules as the universality introduced by Wilson. Let me to finish with the words of Abdus Salam:

   ”Field-theoretic infinities first encountered in Lorentz's computation of electron have persisted in classical electrodynamics for seventy and in quantum electrodynamics for some thirty-five years. These long years of frustration have left in the subject a curious affection for the infinities and a passionate belief that they are an inevitable part of nature; so much so that even the suggestion of a hope that they may after all be circumvented - and finite values for the renormalization constants computed - is considered irrational".