The Famous 1 + 2 + 3 + 4 + 5 + 6 + … + n = -1/12 Demonstration
Introduction
One of the most counterintuitive and fascinating results in mathematics is the equation 1 + 2 + 3 + 4 + 5 + 6 + … + n = -1/12. This equation, known as the infinite series, has been a topic of interest among mathematicians and physicists for centuries. In this article, we will delve into the demonstration of this equation and explore its underlying principles.
The Equation
The equation 1 + 2 + 3 + 4 + 5 + 6 + … + n = -1/12 can be written in a more formal way as:
1 + 2 + 3 + 4 + 5 + 6 + … = Σ(n) = -1/12
Where Σ(n) represents the sum of the first n positive integers.
Demonstration
To demonstrate the equation, we can use the Riemann Zeta function, which is defined as:
ζ(s) = 1 + 2^(-s) + 3^(-s) + 4^(-s) + …
The Zeta function is intimately connected with the distribution of prime numbers, and it has many applications in number theory and physics.
Using the Zeta function, we can rewrite the infinite series as:
1 + 2 + 3 + 4 + 5 + 6 + … = -ζ(-1)
Now, the Zeta function can be analytically continued to a meromorphic function on the entire complex plane, with a single pole at s = 1. This allows us to compute the value of ζ(-1) using the residue theorem.
The Residue Theorem
The residue theorem states that for a function f(z) with a pole of order m at z = a, the residue of f at a is defined as:
Res(f, a) = lim_{z→a} * (z-a)^m * f(z)
Using the residue theorem, we can compute the residue of ζ(s) at s = -1:
Res(ζ, -1) = -1/12
The Final Result
Substituting the result back into the equation, we get:
1 + 2 + 3 + 4 + 5 + 6 + … = -ζ(-1) = -Res(ζ, -1) = -(-1/12) = -1/12
Thus, we have demonstrated the equation 1 + 2 + 3 + 4 + 5 + 6 + … + n = -1/12.
Conclusion
The equation 1 + 2 + 3 + 4 + 5 + 6 + … + n = -1/12 is a remarkable result that has far-reaching implications in mathematics and physics. The demonstration of this equation using the Riemann Zeta function and the residue theorem showcases the beauty and power of mathematical techniques in solving seemingly impossible problems.
