This is perhaps one of the most amusing results I have ever seen in analytica mathematics. It just brought life to the love I had for number theory and analytical mathematics years ago !

As I was reading through some article on string theory, I stumbled upon this result. It just got my mind boggling. I was thrilled. This is it ..

1+2+3+4+5+6+ … infinity = – 1/12

Does that makes sense to you? Actually it did not to me at first sight. How is that even possible? Its crazy. And guess what .. this result is used in major areas of Physics for the foundation formulation. It appears in most of advanced physics texts and is a standard expression used as a corollary in establishing string theory.

string
Page 22 of book String Theory by Joseph Polchinski

But the actual fun lies in the proof of this. I thought it would require something like riemann zeta function or sorts but it is simple, convincing and beautiful. Carefully follow the proof and at the end

I will explain why it is right and wrong at the same time !

S1 = 1 – 1 + 1 – 1 + 1 – 1    ………… ∞

Let us add two S1 series together:

S1  = 1 – 1 + 1 – 1 + 1 – 1     ………… ∞
S1  =   + 1 – 1 + 1 – 1 + 1 – 1 ………… ∞

——————-

2 S1 = 1

2 S1 = 1 => S1 = ½          ——— (1)

S2 = 1 – 2 + 3 – 4 + 5 – 6 +   ………… ∞

We will do similar adding and we get:

S2 = 1 – 2 + 3 – 4 + 5 – 6     ………… ∞
S2 =    + 1 – 2 + 3 – 4 + 5 – 6 ………… ∞

——————-

2 S2 = 1 – 1 + 1 – 1 + 1 ………… ∞

2 S2  = ½     ————— using (1)

=> S2 = ¼    ——— (2)

 

S3 = 1 + 2 + 3 + 4 + 5 + 6 +   ………… ∞

We straight away substract series S2 with S3 as follows:

S3     = 1 + 2 + 3 + 4 + 5 + 6     ………… ∞
– S2   =   + 1 – 2 + 3 – 4 + 5 – 6 ………… ∞

————————————————————
S3 – S2 = 4 + 8 + 12 + 16 + 20 ………… ∞
S3 – ¼ = 4 (S3)

=> S3 = – 1/12

=> 1+2+3+4+5………… ∞ = -1/12

This is a very surprising result. You can find a number of ways to prove it wrong. At least a number of ways to prove the above method wrong. That is okay because there are many ways to prove this and I could also find many ways to prove them wrong. In theory the above result and its interpretation is wrong. When I look at it from numbers point of view I see that  ∞ + 1 is not = ∞. It should be ∞+1 . Just because we are taking infinity with its physical value of something that cannot be computed, so add one to it and it still cannot be computed, this surprising result is showing up. Here we are considering a physical value, a value that cannot be computed, so that the finite value added to ∞ cannot be computed too.

Mathematically, specially for application to the string theory, this result (I believe as of now) is true in the form of :

infinity

And I find it amusing. Its the same feeling I got when my math teacher told me that there is actually a number whose square is -1 !!

I kept thinking all day, but I could not find any intuitive way to represent this result. Neither could I find any explanation online. It looks like some sort of mistake or something, but I am sure it is not. This application is not something so much of part of mathematics as it is part of Physics. This perhaps explain the difference between mathematics itself and mathematics applied to physics I guess and it often lacks definitive answers to a lot of strange anomalies. Cheers !!

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