The sum of all positive integers is... a negative fraction!!!

[z3ro]

 ∞
Σ      n  = 1 + 2 + 3 + 4 + ...    =  - 1 / 12
 n=1


This is awesome. 

[Deque]

Here is a proof:

https://www.youtube.com/watch?v=w-I6XTVZXww

[N3mesis]

math hack    Thats cool.


another one:


.9999repeating = 1


9.999... = 10x


9.999... -  0.999... = 10x - x


9 = 9x


1 = x

[z3ro]

Here is a proof:

https://www.youtube.com/watch?v=w-I6XTVZXww

I.. don't really like the mathematics in this video.

Although those three principles can show those series have the sums listed above, they are contradictory.
   
Here's the proof. Consider that last statement.  From

    1+2+3+4+5+6+....=-1/12
   
subtract itself shifted over

    0+1+2+3+4+5+6+....=-1/12

to conclude

    1+1+1+1+1+1+....=0.

Subtract itself shifted over

    0+1+1+1+1+1+...=0

to conclude

    1=0,

a contradiction.

You cannot use arguments like this to show that the sum of the positive integers is -1/12 since you can use them to show that sum is anything you like.

Now it's true that , but that's an analytic continuation of the zeta function.  It cannot be evaluated by the method described above.  See Numberphile's second video for details.

https://www.youtube.com/watch?v=E-d9mgo8FGk

and

http://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B%AF

[parad0x]

I ak really shocked by seeing this. I mean you don't have something known as common sense? ?  How can the sum of all positive numbers  can be a negative number and that too a negative fraction.  Come on guys, almost anything can be proved by abusing something.

[Deque]

I ak really shocked by seeing this. I mean you don't have something known as common sense? ?  How can the sum of all positive numbers  can be a negative number and that too a negative fraction.  Come on guys, almost anything can be proved by abusing something.

You don't understand. This is not one of these math tricks. This is an actual proof and the result is used in physics. Successfully.
Not everything in math just fits your intuition.

[z3ro]

I ak really shocked by seeing this. I mean you don't have something known as common sense? ?  How can the sum of all positive numbers  can be a negative number and that too a negative fraction.  Come on guys, almost anything can be proved by abusing something.

Common sense? 

Can a particle be at two places at the same time? Damn! Makes no sense. But still... electrons among many others always do that.

Quantum mechanics makes no sense... but that's just the way it is.

And yep. That result is used in many areas of physics. And guess what... IT WORKS! 

[Phenom]

Wow , modern math....So your telling me that 1+2+3+4+5+6...= A negative fraction? I need to do first grade all over again....

[z3ro]

This result is counter intuitive... but real. 

The thing is, this analytic continuation makes both mathematical and physical sense -- physics experiments have confirmed it. For example, the well-known Casimir effect, which is the electrical attraction of two parallel, uncharged metal plates due to quantum effects, is calculated to involve the Zeta function ζ(-3), which by similar reasoning equals 1/120, and the predicted Casimir force has been verified by experiments. That is

1 + 2³ + 3³ + 4³ + ... = ζ(-3) = 1/120

is experimentally verified.
https://en.wikipedia.org/wiki/Casimir_force

Wikipedia has an entry on this sum, including a brief one on its relationship to string theory, and this wonderful excerpt from a letter from the Indian genius Srinivasa Ramanujan's to the mathematician who eventually brought him to Englaand, G. H. Hardy:
"Dear Sir, I am very much gratified on perusing your letter of the 8th February 1913. I was expecting a reply from you similar to the one which a Mathematics Professor at London wrote asking me to study carefully Bromwich's Infinite Series and not fall into the pitfalls of divergent series. … I told him that the sum of an infinite number of terms of the series: 1 + 2 + 3 + 4 + · · · = −1/12 under my theory. If I tell you this you will at once point out to me the lunatic asylum as my goal. I dilate on this simply to convince you that you will not be able to follow my methods of proof if I indicate the lines on which I proceed in a single letter. …"

https://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B%AF

[Phenom]

^ +1