Ramanujan's Summation || What is the sum of all counting numbers?

    Well, we all know that the sum of first n natural numbers is n(n+1)/2. Fine! But the mystery is, what is the sum of all counting numbers? Well, this is no longer a mystery, as the great Indian mathematician Srinivasa Ramanujan gave the sum for the same.

                              Srinivasa Ramanujan

         He suggested that the sum of all counting numbers (all natural numbers upto ♾️) i.e. 1+2+3+4+5+6+7+••• upto ♾️ is -1/12. If you still feel it difficult to believe, if you think, how all positive numbers sum up to negative number? Well, don't think too much, keep reading my article to unreveal all the mysteries.

         The following two series are the backbone of our proof for the sum:

1. 1-1+1-1+1-1+1-1+1-1+•••• = 1/2
2. 1-2+3-4+5-6+7-8+9-10+••• = 1/4

         Well we have already discussed about the first series, that is Grandi's series. The crazy one! In fact I have one whole article on this little beauty called Grandi's series. Check out that. There we proved; the series 1-1+1-1+1-1+1-1+••••upto ♾️ sum up to 1/2. 

          Now let's talk about second series;

B = 1-2+3-4+5-6+7-8+•••• 

          let's have some fun with this series; Add B series to itself, but wait, by using small trick. Just shift the series by a unit position. So, the series is now,

           B= 0+1-2+3-4+5-6+7-8+•••• ( adding zero doesn't affect anything)

So, double B, we get,

     B+B = 1-2+3-4+5-6+7-8+•••

              +0+1-2+3-4+5-6+7-8+••••

So, 2B = 1-1+1-1+1-1+1-1+••••

The series on RHS is the series that we have talked already.( Grandi's series) Which sum up to 1/2.

Hence, 2B = 1/2 

       So, B = 1/4.

Here we go, we got this. This crazy series don't have any crazy name. 

        Now coming for original claim, that is:

        1+2+3+4+5+6•••= -1/12.

So, consider S = 1+2+3+4+5+6+7+••••

Simply subtract B from S.

We get,

      S - B = 1+2+3+4+5+6+7+••••

                -(1-2+3-4+5-6+7-8+••••)

      S - B = 1+2+3+4+5+6+7+••••

                  -1+2-3+4-5+6-7+••••

      S - B = 0+4+0+8+0+12+0+16+0+••••

                = 4+8+12+16+••••

                = 4(1+2+3+4+5+6+••••)

You see, this new series is exactly four times for original series S.

So, S - B = 4 S

       3S = - B

       3S = -1/4.  ( Since B = 1/4 )

        S = -1/12.

Hurrah! We got this. Our sum is negative. That's why I always say, Mathematics is just Awesome, it is just marvelous! When I read of this first time, I was really shocked, just like you. 

        It seems, this is some weird illusion or mathematicians are playing crazily with us. But, let me tell you this is very valid and important sum. This sum has remarkable use specially in string theory. It is very useful in Quantum field theory and  in some general physics also.

Thank you for reading.