The sum of all natural numbers (1 + 2 + 3 + ...) to infinity is not a finite number; it diverges. However, in the context of certain mathematical techniques, particularly in the realm of analytic continuation of the Riemann zeta function, the sum is assigned the value -1/12. This result is not a sum in the traditional sense but arises from extending the definition of summation to include infinite series that would normally be considered divergent.
Understanding the Divergence
As stated in multiple sources (Wikipedia, Math.ucr.edu, Quora), the sum 1 + 2 + 3 + 4 + ... increases without bound. There is no finite number it approaches; it diverges to infinity. Any attempt to assign a finite value using conventional methods of summation will fail.
The -1/12 Result: Analytic Continuation
The result of -1/12 is obtained through advanced mathematical techniques involving the Riemann zeta function. This function is defined for complex numbers with a real part greater than 1. However, using analytic continuation—a method of extending the domain of a function—mathematicians can assign a value to the function at s = -1. The Riemann zeta function at s = -1 corresponds to the sum 1 + 2 + 3 + ..., and the analytic continuation yields the value -1/12. This is mentioned in several references (ResearchGate, Reddit, Medium). It's crucial to understand this value does not represent the sum in the traditional sense of adding numbers.
- Traditional Summation: Diverges to infinity.
- Analytic Continuation: Assigns the value -1/12, but this is a result of extending the definition of summation beyond its conventional meaning.
This -1/12 result has implications in various areas of physics, particularly string theory, but it's vital to remember its non-conventional nature. The sum itself doesn't actually equal -1/12 in the typical sense of addition. The result stems from a sophisticated mathematical process applied to a function related to the sum, not the sum directly.
