How 1+2+3+...+∞ = -1/12 ?

The statement that 1+2+3+...+∞ = -1/12 is known as the Ramanujan summation, after the Indian mathematician Srinivasa Ramanujan. It is a controversial result, and there is no consensus among mathematicians about whether it is actually true.

The sum of all the natural numbers, 1+2+3+...+∞, is a divergent series. This means that it does not have a finite sum in the traditional sense. However, there are ways to assign a value to this series, and one of the most popular methods is the Ramanujan summation.

It is a way of assigning a value to divergent series by taking into account their analytic continuation. Analytic continuation is a way of extending the definition of a function to complex numbers, even if the function is originally defined only for real numbers.

The Ramanujan summation of 1+2+3+...+∞ is equal to -1/12. This may seem strange at first, but there is actually a way to see why this is the case.

One way to look at it is to consider the alternating series 1-2+3-4+...+∞. This series is also divergent, but it is much easier to work with. In fact, the sum of this series is equal to 1/2.

The Ramanujan summation of 1+2+3+...+∞ can be derived from the sum of the alternating series by using a technique called the Euler-Maclaurin summation formula. This formula allows us to express the sum of an alternating series as a combination of the terms of the series and their derivatives.

Using the Euler-Maclaurin summation formula, we can show that the Ramanujan summation of 1+2+3+...+∞ is equal to -1/12.

Here is a more detailed explanation of the proof:

The Euler-Maclaurin summation formula states that the sum of the alternating series 1-2+3-4+...+∞ is equal to:

1/2 - 1/6 + 1/12 - ... = 1/2

We can extend this formula to the sum of the series 1+2+3+...+∞ by adding a constant term to each term in the series. This gives us the following formula:

C + 1 + 2 + 3 + ... = -1/12

The constant C can be determined by considering the fact that the sum of the series 1+2+3+...+∞ should be equal to zero when the number of terms is even. This gives us the following equation:

C + 1 + 2 + ... + n = 0

Solving this equation for C, we get C = -1/2.

Substituting this value of C into the first formula, we get the following result:

-1/2 + 1 + 2 + 3 + ... = -1/12

Therefore, the Ramanujan summation of 1+2+3+...+∞ is equal to -1/12.

Whoosh!!, That was a lot a maths!!

Then, let's see where Ramanujan's summation is used.

Applications of the Ramanujan Summation

The Ramanujan summation has been used in a variety of fields of science, including:

Number theory: The Ramanujan summation has been used to study the Riemann zeta function, which is a function that is closely related to the distribution of prime numbers. The Riemann zeta function is also a divergent series, but the Ramanujan summation can be used to assign a value to it.
Physics: The Ramanujan summation has been used to study the Casimir effect, which is a force that arises between two uncharged parallel plates in a vacuum. The Casimir effect is also related to divergent series, and the Ramanujan summation can be used to calculate the magnitude of the force.
String theory: The Ramanujan summation has been used to study string theory, which is a theory of elementary particles that is based on one-dimensional objects called strings. String theory is also related to divergent series, and the Ramanujan summation can be used to calculate some of the properties of strings.

Other Applications

In addition to the applications mentioned above, the Ramanujan summation has also been used in other fields of science, including:

Quantum mechanics: The Ramanujan summation has been used to study the energy levels of atoms and molecules.
Statistical mechanics: The Ramanujan summation has been used to study the behavior of systems of particles at a macroscopic level.
Financial mathematics: The Ramanujan summation has been used to study the behavior of financial markets.