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Sum of all positive numbers equals a negative fraction?

 

Sum of all positive numbers equals a negative fraction?  

 

 

Let me ask you a question. What do you think the sum of natural numbers up to infinity is?  

 1+2+3+4+5+6+...∞ =?

 I know your answer is infinity. What if I told you that the answer was some negative number? You would probably beat me up and file a case against me. But wait, prepare your army before coming for me, for I am not alone in this thing, and think twice about who to recruit in your team, for I have several physicists and mathematicians like Ramanujan in my team.  

 

Ramanujan, in his famous letter to G.H. Hardy, mentioned these infinite sums:


 

 

Euler-Riemann zeta function for negative value of ‘s’ also gives us the similar infinite sums.


For s= -1,

 

 Many say that it is just the value of Euler-Riemann’s zeta function at s = -1, which we get after the analytic continuation of the function, but the most bizarre thing about this sum, and which compels you to believe on it, is that it is used in String theory and many other fields of physics; it is critical in understanding the Casimir force in quantum electrodynamics.

 Now, you must be thinking how we even get that number. Fasten your seat belt and let us all be ready for a mathematical ride of fantasy, of imagination, where the word manipulation doesn’t exist. 

 

Let us consider some infinite sums,

 S₁=1-1+1-1+1-1+1.....

  S₂ = 1-2+3-4+5-6....


    s=1+2+3+4+5+6.....


What we know about the first sum is that if we stop at even number of terms, the result will be zero, and if we stop at odd number of terms, the result will be ‘1’, but Infinity is a beautiful concept of never stopping, never ending. Therefore, we, intuitively, take an average of 1 and 0 to attach a real value to the sum.  

              S₁=1-1+1-1+1-1+1.........=1∕2

 

There are a lot of other ways to come across this strange looking result, which we can discuss some other day. Now, our next step will be to attach a real value to the second sum and finally get the value of final sum.

                            S₂ = 1-2+3-4+5-6....

          S₂ = 1-2+3-4+5-6....

           2S₂ = 1-1+1-1+1-1+1........ 

            or, 2S₂ = 1/2

             ∴S₂ = 1/4

Similarly, 

                  s =1 + 2 + 3 + 4 + 5 + 6.......

                   -(s₂ =1 - 2 + 3 - 4 + 5 - 6 ........) 

                _________________________

                      s - s₂ = 4 + 8 + 12......


                or, s - (1/4) =4(1 + 2 + 3 + 4 +.....)

                or,  s- (1/4) = 4s

                or,  3s = -1/4

                 ∴ s = -1/12


This result is still debatable, but if you asked me my opinion on this, I would say 1+2+3+..... equals infinity and there must be a way to assign real values to infinities. If you take a closer look at values of Riemann zeta function for negative integer values of ‘s’, you will find that for odd negative integer values of s, the function gives fractions having multiple of 3 in its denominator, and for even negative integer values of s, it gives simply 0. And, there are many other things I noticed during my research on infinity, which I cannot discuss here. However, feel free to contact me if you want to know more.  

 

The beauty of science is that it keeps changing and accepts things that were thought impossible once. Several centuries back, the idea of heliocentric solar system was so controversial that the Catholic Church classified it as a heresy, and warned Galileo to abandon it. But we know what the truth is in today’s time, courtesy of Nicolaus Copernicus who proved that the earth revolves around the sun and not the other way around in 1543. So, before drawing any conclusion, let us all wait for our mathematical Copernicus to come and put an end to this debate forever. 

 

 

- Md Asfaque ( mdasfaque59@gmail.com) 



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