The Ramanujan Summation: 1 + 2 + 3 + ⋯ + ∞ = -1/12.
When I was first told by my friend about this, I was just blown away. I thought he is joking around, but then I got to know that this is real. Then I thought that most of the scientists must have rejected this, but no. I was wrong, it is even used in higher physics.
This was and is one of the most mind blowing mathematical facts that I came to know.
It is amazing how each positive numbers are added and the result you get is negative.
Here's the prove which is even more interesting.
The prove given below is just for fun and it is not valid. So if you want to enjoy then only read and if you are a mathematician then please don't read.
Consider
S1= 1-1+1-1+1-1+1-1…..
Now, this sum should be 0 or 1 based on number of natural numbers taken. If infinite numbers are even, S1=0, if odd then S1=1. But, Riemann zeta function gives it a value of ½. Mathematical community too agrees that the sum is ½.
Let
S2=1-2+3-4+5-6+7…..
So, S2=1-2+3-4+5-6+7-8+9…..
S2= 1-2+3-4+5-6+7-8……. I have shifted RHS by a unit position
+2S2=1-1+1-1+1-1+1…..
Hence, 2S2=S1
Therefore, S2=1/4
Let’s come back to our sum of infinite numbers.
S=1+2+3+4+5+6+7+8+9…..
S2=1-2+3-4+5-6+7-8+9….
S-S2=4+8+12+16+20…..
Hence,
S-S2=4(1+2+3+4+5+6+7+8….)
S-S2=4S
So, -S2=3S
And, S = -S2/3 = -1/12
This shocking result is not known to many non-mathematicians. Number-theorists call it “One of the most remarkable formulae in science”. This summation is a secret of mathematics kept away from layman. Further, it is interesting to know ‘S’=-1/12 has been used to derive the equations in “string theory”, quantum field theory and in some complex analytics.
EDIT: here is a new proof for all mathematicians:
I have given the following propositions. If you want a prove of that the please comment.
Proposition 1.
(2.1)
Proposition 2.
(2.2)
We define the function Sn and Hn (x) as follows.
(2.3)
(2.4)
Then we define the following symbols.
(2.5)
(2.6)
(2.7)
(2.8)
Then we have the following propositions.
(2.9)
(2.10)
(2.11)
The double quotes mean the analytic continuation of the sum of natural numbers.
The traditional sum diverges for the infinite terms. On the other hand, the new “sum” is equal to the traditional sum for the finite term. In addition, the “sum” converges on -1/12 for the infinite term.