I am sure you must have heard about various kinds of series during your high school. Remembering the formulae for their summation without knowing what practical application could such series have was boring, right? Not anymore. There are various interesting sequences which have surprising results. They have been used by researchers and scholars to develop various theories. One such among them is the ‘Ramanujan Summation’.
The results of the summation are so astonishing that it makes us question what we have been taught. For instance, if we have to find the sum of the first 50 natural numbers i.e. 1 + 2 + 3 + … + 50, we would use formula for the sum of n numbers of an arithmetic progression or directly the formula for the sum of first n natural numbers i.e. n(n+1)/2 So, we would get 50 * (51/2) = 1275 Now, what if we have to find the sum of all the natural numbers till infinity? Would it be a finite number? Would it be a positive number? You might think it would certainly be a positive number and probably the answer would also be infinity . But unfortunately, that is wrong. Can you believe the answer to the question is not a positive number? No, right? However, the summation results in -1/12 .
Srinivasa Ramanujan, who we today call ‘The Man Who Knew Infinity’, was among the first to give this summation and hence the name. Ramanujan was a natural genius. Ramanujan Summation essentially is a property of partial sums. The above summation also involves Euler-Maclaurin summation formula together with the correction rule using Bernoulli numbers.
Wondering how we got a rational number, let alone being negative?
Let us try to understand the proof of the above sum. However, it must be noted that the infinity that we talk about in this article is a countable infinite set, which is slightly different from the normal infinity that we know. The assumption is taken because only then would we be able to apply the mathematical properties.
We need to find,
Let us take,
What would result in? Either 0 or 1, right? Again, wrong. This series is divergent, a series where the infinite sequence of the partial sums of series do not have a finite limit. This type of divergent series is called Grandi’s divergent series. An interesting fact to note here is that, the result to this series of X can also be found using the geometric series i.e. the sum of infinite numbers of a Geometric Progression, only after making certain adjustments.
It seems like we are forcefully trying to arrive at the result of X = 1/2, but it is indeed true. The method we have used to get the value of X is not absolutely correct. We have to know much complex concepts like that of the Zeta Function to understand the same. For now, we take it to be 1/2
Now, let us consider,
Now, we have two equations,
It seems like some kind of magic. Doesn’t it? Well, that is the beauty of Mathematics. For years, this summation was highly debated. However, the result has been used in various theories of Physics. It has been used to derive equations in supersymmetric string theory. It was helpful in other areas of general Physics like to find the solution to the Casimir Effect, an effect arising from quantum theory of electromagnetic radiation.
One more interesting fact is that Ramanujan not only defined the sum of infinite natural numbers but he also defined the sum of the square of the infinite natural numbers and the sum of the cubes of infinite natural numbers i.e. 1^2 + 2^2 + 3^2 + … and 1^3 + 2^3 + 3^3 +4^3 + …
By now, you might have assumed that the answer to these series is also going to be something weird. Yes, you are right. The above sums result in 0 and 1/120respectively.
The above two sums can be proved using Riemann Integrals of real valued functions and hence it requires a deep knowledge about when the integral can be used.
At first, it might seem that Ramanujan’s work does not seem logical, and this is what mathematicians back then thought and hence neglected his work. However, after a few years, mathematicians got to know how the sums have been arrived at and it seems that he tried to convey something much deeper than what we know. Ramanujan would always be remembered for his great work and his contribution despite facing hardships and not having formal knowledge of advanced mathematics.
We will try to cover the proof of the remaining two summations in upcoming blogs after talking about Riemann Integrals. I hope you found this one insightful!