Have you ever wondered about divergent series? These are the series which do not converge to a finite sum. Take for instance the following infinite series:
We know that this series converges to 2 at infinity. Such series are known as convergent series. The series which we are interested in, however, are the ones called divergent series.
N.H. Abel wrote
❝The divergent series are the invention of the devil, and it is a shame to base on them any demonstration whatsoever.❞
Divergent series cannot be summed by usual methods but they can be ❛summed❜ (read: associated) to a number using other rigorous methods.
One famous example of a divergent series is the sum of all natural numbers upto infinity.
S = 1+2+3+4+5 ...
This series is particularly important because it has applications in other fields like Bosonic String Theory, Quantum Field Theory et cetera.
There are many methods to ❛sum❜ this series which require different amounts of rigor. I'm going to discuss two methods, the one using Cesaro Summation of Grandi's Series and Euler's Method. Other methods include Zeta Function Regularization, Ramanujan Summation, Exponential Regulation Method et cetera.
The first method uses the following two series, the first one is called Grandi's Series
S0 = 1-1+1-1+1-1 ...
S1 = 1-2+3-4+5...
Now, the Grandi's series is mildly divergent as it can either be summed to 1 or 0. The graph of Grandi's series is a zig-zag 'oscillating' between 0 and 1 which can be thought as a straight line parallel to X-axis at y=1/2 or in simple terms an average of the two possibilities i.e. (0+1)/2 = 1/2. Cesaro Summation also sums Grandi's series to 1/2.
S0 = 1-1+1-1+1-1 ... = 1/2
S1 = 1-2+3-4+5 ...
+S1 = +1-2+3-4+5 ...
2S1 = 1-1+1-1+1-1 ... = S0 = 1/2
Therefore, S1 = 1/4.
S-S1 = [1+2+3+4 ...] - [1-2+3-4 ...]
As we can see all the odd terms will cancel out and even terms will add up resulting in:
S-S1 = 4+8+12+16 ... = 4 (1+2+3+4 ...) = 4S
Now, -3S = S1 = 1/4.
Therefore, S= -1/12 or 1+2+3+4 ... = -1/12.
This method uses Taylor series expansion of 1/(1+x)2 to prove that S1 or 1-2+3-4 ... is equal to 1/4.
1/(1+x)2 = 1-2x+3x2-4x3 ...
Putting x=1 in the above equation reduces it to the following:
1/22 = 1-2+3-4 ... = S1
Now that we have proved S1=1/4 we can easily continue as above to prove S=-1/12.
It might seem bizarre to associate -1/12 to 1+2+3+4 ... but it has serious physical implications and is a well established fact with other rigorous proofs.
Here is an excerpt from the letter of S. Ramanujan to G.H. Hardy on the topic:
❝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.❞