Ramanujan Paradox Proof - Ramanujan Summation - Sum of all natural numbers by Ramanujan - Ramanujan

Here is the proof of Ramanujan infinite series of sum of all natural numbers. This is also called as the Ramanujan Paradox and Ramanujan Summation.
In this video you will also learn about the life of Indian Genius Mathematician Srinivasa Ramanujan.
The video proves how the sum of all natural numbers up to infinity is -1/12 and also the applications of the Ramanujan Paradox (Ramanujan Summation) in physics.
Do you also think that the sum of the series that you are seeing here, is wrong?
After all how the sum of an infinite series having all the natural numbers which are all positive can be a negative number?
At first it looks impossible but let me tell you that this is actually correct.
The sum of this series was found by a genius mathematician named Ramanujan. So this is also called as the Ramanujan Paradox.
Let me tell you that, this is not just in theory, It actually happens and its proof finds application in many physics problems like Casimir Effect.
It is also famous for its use in Bosonic String theory where it is used to calculate the dimensionality of the universe.
So here I am going to prove how the sum of this infinite series is equal to -1/12.
After the proof I'll tell you some unbelievable facts related to the Srinivasa Ramanujan- the man who knew infinity.
Ramanujan was an Indian mathematician who had no formal training in pure mathematics but made great contribution to mathematics.
The famous mathematician G.H. Hardy came up with a scale to calculate the mathematical ability of mathematicians. The scale was from level 0 to 100.
Hardy put himself at level 25. The the great German mathematician Hilbert at 80 and Ramanujan at 100.
Ramanujan lived for very short period of time and died at the age of just 32 years.
But in his short period of life he established more modular equations than Jacobi, Gauss, Eulers and other great mathematicians combined.
It took the whole life of many mathematicians just to prove Ramanujan's one percent of work that was found in his notebook.
He even provided solutions to many mathematical problems which were considered unsolvable at that time.
Ramanujan found out thousands of equations but most of them are still unproven.

There is much to say about Ramanujan but this video was actually about proving the sum of the infinite series.
I hope you found the information about Genius mathematician Ramanujan interesting.

[RickLambert]

That was the simplest explanation I’ve seen yet. Well done. It’s a shame he died so young.

[alkispiskas]

There's a big problem to start with: We cannot talk about the end of an infinite series, i.e. if the total number of items in a series is even or odd, since **infinity by definition has no end**.
Actually, it has not even a start. So starting with 1 or whatever else has no meaning either.

[soumyadeepdash]

How can we take two equations as a same variable (s1) as their sum is different ?

[panchananpramanik]

It ia mathematical joke . When sum of a infinite polynomial is divergent or convergent, infinity is valid . But when oscilitory then it has no solution .

[utkarshgupta]

How is this proof even right?
S1 is having two value and the same time

Than when we shift S2 series by one place than after S2+S2 first 1 in starting and last number that will be a big infinitely lastge number will be left after cancelling all 1 and -1
Than means S2= ( 1 + inf. Large number)/2

So much blunder but still it is said right. How sir please explain where am I going wrong?

[sorindom]

"The Man Who Knew Infinity" movie , The story of the life and academic career of the pioneer Indian mathematician, Srinivasa Ramanujan, and his friendship with his mentor, Professor G.H. Hardy ...

[ns]

So far so good Ramanujan was true legend and more of all the staunch bhakt of Bgagvati. However my question here is that why we had taken the substraction of the no. 1 till infinity? Moreover in s4 why did we shift the no.s to 1 place right for substracting them from s3? If we'd not followed this the outcome would'n different? There are more questions but wanted to start with the basic first. Thanks.

[Clodd]

This makes no sense for some reasons. First, you keep increasing so how can the result be lower? If you add 1 to infinity (infinity + 1) it will simply expand not shrink. Secondly, why is a subtraction used on something that keeps increasing? You could simply use the odd numbers for the odd series (2 + 4 + 6 + etc).
If I keep adding more and more apples in my basket, at some point I would have less than half apple? How?

[Arv.-]

Stop killing science ... if you can not do good, please do not harm mathematics. Stop talking nonsense.

[poojik]

If he were alive for more years, I think that he would have proved the absolute reality...

[محمدحارثبھائی]

But how can S1 be equal to 0 ( when the series has EVEN no. of terms ) and be equal to 1 ( when ODD ) at the same time?

in the very beginning

[emochies]

🤔s¹ maybe fine but, how can both s² have two different value when they have same number. First s¹ is natural number when second is whole? How can they be added?

[drvasuslectures]

At 3.57 you write conveniently as 1-1+1-1+1-1+1. How can it be? It will be 1+7
What did you do with that last 7. Check it

[augustjologs]

So many mathematical mumbo-jumbo, hocus-pocus steps here but conveniently violated the basic mathematical axiom of Addition Property of Equality (and by extension, Multiplication Property of Equality and all other mathematical properties of equalities). 1. The APE's use is limited to finite NUMBERS. Once you include infinity in the equation, you cannot use APE, otherwise you will get weird results, like this Ramanujan Summation. 2. Infinity is a CONCEPT, it is not a number. Hence it cannot be subjected to mathematical operations. How 'mathematicians' failed to see these is beyond me (I'm not a mathematician).

[ronbally]

I have no problem with the -1/12, but the equal sign requires the notion of a regularized sum. Stating that such sums are equal to a value, is only true given a method to assign a value, like Ramanujan Summation, or running average, or …. But that is not a “normal” summation.
I also believe that what you are showing is NOT Ramanujan Summation.
All of this is done quite well and complete by Mathologer, which I would recommend to view.

[nooruddinbaqual]

This is a mathematical gimmick and to prove it, you are one time taking even number of terms and then taking odd number of terms.In our school days We were proving 4=5.We were taking recourse to the fact that square root of a perfect square number is both Plus and and Minus.
I love Ramanujan for his Partition Theory of Numbers.

[O-Kyklop]

There is something quite wrong with this paradox. If Ramanujan is working with numbers he is not working with the Infinite, he can't be working with the infinite because the Infinite has no numbers.
He can only work with numbers after finding the frame of the numbers.
You can't just write numbers if you don't know where are you placing them.

[keananspach]

Not sure why it is ok to shift a series one place to add or subtract from another. Why not 2 places? Why not 3?

[localboygaming]

How can you sum s1 and s1 if they have diff values?

[walidcless]

As a person who loved Math since I was a child, I just can't accept this, my mind can't accept this, complete rubbish if you ask me