Srinivasa Ramanujan, one of the most brilliant mathematicians of the 20th century, introduced a unique method for dealing with divergent series, known as Ramanujan’s summation. Traditional summation techniques fail for these series, making Ramanujan’s approach particularly innovative.

One of the most famous examples of Ramanujan’s summation is his handling of the infinite sum of natural numbers: 1 + 2 + 3 + 4 + . . . Intuitively, this series diverges to infinity.

However, using his summation method, Ramanujan assigned it a surprising and counterintuitive value of −1/12. This result, although not a sum in the conventional sense, is consistent within the framework of Ramanujan’s summation and has significant implications in various fields, including theoretical physics and string theory.

What is Ramanujan’s Infinite Sum?

Ramanujan’s infinite sum, often referred to as the Ramanujan summation, involves the sum of all positive integers:

1 + 2 + 3 + 4 + 5 + . . .

Intuitively, the sum of all positive integers diverges to infinity. However, Ramanujan discovered a way to assign a finite value to this divergent series using techniques from complex analysis and analytic continuation. He found that:

1 + 2 + 3 + 4 + 5 + . . . = −1/12

This result is surprising and counterintuitive, and it’s important to note that it doesn’t mean the sum of all positive integers is literally −1/12​. Instead, this result arises in specific contexts, such as string theory and the study of the Riemann zeta function.

One way to understand this result is through the regularization of the series using the Riemann zeta function, ζ(s), which is defined as:

[Tex]\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}[/Tex]​

For Re(s) > 1, this series converges. However, through analytic continuation, it can be extended to other values of sss. For s = −1, we have:

ζ(−1) = −1/12​

Since the original series 1 + 2 + 3 + 4 + . . . can be formally written as ζ(−1), we get:

1 + 2 + 3 + 4 + 5 + . . . = ζ(−1)

Who was Srinivasan Ramanujan?

Srinivasa Ramanujan (1887-1920) was an Indian mathematician who made substantial contributions to mathematical analysis, number theory, infinite series, and continued fractions. Despite having no formal training in pure mathematics, Ramanujan developed his theories and made remarkable discoveries that have influenced various fields of mathematics.

What is an Infinite Sum?

An infinite sum, also known as an infinite series, is the sum of the terms of an infinite sequence. Mathematically, if a₁, a₂, a₃, . . . is an infinite sequence of numbers, then the infinite sum (or series) is written as:

[Tex]S = a_1 + a_2 + a_3 + \cdots[/Tex]

Mathematical Formulation of Ramanujan’s Infinite Sum

Mathematical formulation of Ramanujan’s infinite sum is affirmed with the help of Reimann Zeta function. Let’s discuss what is Reimann Zeta Function first.

Riemann Zeta Function

The Riemann zeta function ζ(s) is defined for complex numbers sss with Re(s) >1 by the series:

This series converges when Re(s) > 1. However, ζ(s) can be analytically continued to other values of sss except for s = 1, where it has a simple pole.

Evaluation at s = -1

Through analytic continuation, it is possible to evaluate the zeta function at s = −1:

[Tex]\zeta(-1) = \sum_{n=1}^{\infty} [/Tex]

Surprisingly, the value of ζ(−1) is:

ζ(−1) = −1/12

This result assigns a finite value to the divergent series 1 + 2 + 3 + 4 + 5 + . . .

Conclusion

Ramanujan’s infinite sum is formally written as:

1 + 2 + 3 + 4 + 5 + . . . = −1/12

This result is obtained through the analytic continuation of the Riemann zeta function and is used in various advanced areas of mathematics and theoretical physics.

Read More,

• Sum of an Infinite Geometric Progression
• Sum of Infinite Series Formula
• Product of Infinite Series

FAQs on Ramanujan’s Infinite Sum

What is Ramanujan’s Infinite Sum?

Ramanujan’s infinite sum refers to the assignment of a finite value to the divergent series 1 + 2 + 3 + 4 + . . . Using regularization techniques, particularly through the analytic continuation of the Riemann zeta function, this sum is assigned the value −1/12.

How can the sum of all positive integers be −1/12​?

Intuitively, summing all positive integers leads to infinity. However, in the context of analytic continuation and specific regularization techniques, the divergent series 1 + 2 + 3 + 4 + . . . is assigned the finite value −1/12​. This does not mean that the literal sum of these numbers is −1/12​; rather, it is a result from the analytic continuation of the Riemann zeta function ζ(s) evaluated at s = −1.

What is analytic continuation?

Analytic continuation is a technique used to extend the domain of a given function beyond its original domain. For the Riemann zeta function, this involves extending its definition from Re(s) > 1 to other values of s. This allows the evaluation of the zeta function at points where the original series does not converge, such as s = −1.

Who was Srinivasa Ramanujan?

Srinivasa Ramanujan (1887-1920) was an Indian mathematician who made substantial contributions to mathematical analysis, number theory, infinite series, and continued fractions.

How is Ramanujan’s sum used in string theory?

In string theory, the value −1/12 is used in the calculation of the zero-point energy of a string.