The sum of 1-2+3-4+5 , and divergence

[asd1249jf]

The sum of 1-2+3-4+5..., and divergence

The sum of 1-2+3-4+5..., which can be written as

Diverges for m = infinity, yet there are postulates that this is equal to $$\frac{1}{4}$$.

First, I don't understand how you can obtain a fraction out of a natural numbers if they are consecutively added, regardless of if the number is positive or negative.

Second, I don't understand how this is equal to 1/4, when the infinite series diverges. Can anyone help me understand this postulate?

 

[SiddharthM]

there was a another post dealing with what might be a similar issue - the harmonic series converging to -1/12 or something like that. (search for ramanujan summation)

The limit of partial sums here does not exist as the even partial sums goto negative infinity and the odd partial sums go2 infinity, so by using the theory of infinite series one typically learns in calculus/analysis classes, this definitely diverges.

 

[Gib Z]

Yes the thread Sid talks about is in fact my own. There are different methods of summation, and the most common one is defined as the limit of partial sums, which in this case it certainly diverges. However they have other methods of 'summation' which assign values to series that are divergent. Sometimes in physical instances, one needs to sum all the natural numbers, but expects a finite answer on finite grounds! It turns out that Ramanajan summation, a different method, assigns a value to it, -1/12, and it actually turns out to be a very nice answer within string theory because of the number of dimensions.

Anyway, There are other 'alternative' methods of summation, and some are actually quite good because for convergent series, they yield the same value as the regular limit of partial sums definition. So read up on different summation methods, some come to mind so as Caesaro, Ramanajan or Pade summations.

 

[mathman]

Formal approach:
1-2x+3x^2-4x^3+...=d/dx(-1+x-x^2+x^3-x^4+...)=d/dx(-1/(1+x))=1/(1+x)^2.

The above formalism holds for |x|<1. However, using analytic continuation (pole is at -1) we can evaluate it at x=1 and get 1/4.

 

[AlphaNumeric2]

First, I don't understand how you can obtain a fraction out of a natural numbers if they are consecutively added, regardless of if the number is positive or negative.

Second, I don't understand how this is equal to 1/4, when the infinite series diverges. Can anyone help me understand this postulate?

In string theory, there's a summation of modes which amounts to 1+2+3+4+... Using regularization, it's computed to be -1/12.

That, just like your series, is computed by taking a function which 'works' for a given range and making it valid within the entire complex plane (analytic continuation). Here and there there will be poles, where the quantity is infinite but even there (like Zeta(-1) = 1+2+3+... = infinity) you can compute results by considering the value of the pole at that point.

Often, it turns out that the motivation for your summation (like the modes in string theory) is actually a little more elaborate than your initial naive assumptions and you can make your use of analytic continuation and poles rigorous.

 

[Count Iblis]

Borel resummation:

$$\sum_{n=1}^{\infty}(-1)^{n+1}n=\sum_{n=1}^{\infty}(-1)^{n+1}\frac{n}{n!}\int_{0}^{\infty}dx x^{n}\exp\left(-x\right)=\int_{0}^{\infty}dx x\exp\left(-x\right)\sum_{n=0}^{\infty}(-1)^{n}\frac{x^{n}}{n!}=\int_{0}^{\infty}dx x\exp\left(-2x\right)=\frac{1}{4}$$

 

[asd1249jf]

I don't understand, the formal approaches I've seen so far may indeed give 1/4, but realistically, that is impossible. How do you obtain a fraction out of a continuous addition/substraction of pure integers? Analytically, it's impossible.

 

[Gib Z]

The alternative forms of summation are summing the terms in different ways than just adding and subtracting, it a way its giving the average value of the partial sums, even if the partial sums don't converge.

 

[leon1127]

I don't understand, the formal approaches I've seen so far may indeed give 1/4, but realistically, that is impossible. How do you obtain a fraction out of a continuous addition/substraction of pure integers? Analytically, it's impossible.

This never makes any sense.
i.e a finite sequence {1,2,3,4,5,6,7,..., n} are integers for all n integers. If we let n goes to infinity, why is infinity not integer? Again, infinite is a concept. We force it to have a meaning that is not possible in reality. Nothing is infinitely small or large.