# What is the sum of all the natural numbers?

The sum of all natural numbers is not a finite value. It is technically infinite. When we say the sum of all natural numbers, we refer to the series:

1 + 2 + 3 + 4 + 5 + 6 + …

As you can see, there is no end to this series; it goes on forever. Hence, the sum of all natural numbers is said to be infinity (∞).

However, there is a concept in mathematics known as “divergent series,” and this particular series, 1 + 2 + 3 + 4 + 5 + 6 + …, is an example of a divergent series. In the context of mathematical analysis, it’s not assigned a finite value because it doesn’t converge to a specific sum.

The sum of the series 1 + 2 + 3 + 4 + 5 + 6 + … is not -1/12. The idea that this series sums to -1/12 is a common misconception that arose from a mathematical regularization technique used in certain areas of theoretical physics, specifically in the field of string theory.

In the context of mathematical analysis and traditional number theory, this series is divergent, meaning it does not have a finite sum. The sum of positive natural numbers growing infinitely is not assigned a finite value.

However, in certain areas of theoretical physics, specifically in string theory and quantum field theory, the concept of “zeta function regularization” is employed to assign a value to certain divergent series. When this regularization technique is applied to the series 1 + 2 + 3 + 4 + 5 + 6 + …, it yields a value of -1/12. This result has been used in some mathematical contexts within theoretical physics but does not imply that the sum is truly -1/12 in the traditional mathematical sense.

It’s essential to note that the sum of all natural numbers being -1/12 is a specialized interpretation within specific mathematical contexts and is not the standard understanding in traditional mathematics. In most practical mathematical applications, the series remains divergent, and the sum is considered to be infinity (∞).

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This is a great post. I didn’t know the usage of it in String Theory and QFT.
Keep it up !!

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It is very easy. The answer will be

Imagine,
S = 1+2+3+....
–›S = 1+(2+3+4)+(5+6+7)+(8+9+10)+......
–›S = 1+9+18+27+......
–›S = 1+9(1+2+3+.....)
–›S = 1+9S
–›S - 9S = 1
–› -8S = 1
\color{red}{–›S = - \frac{1}{8}}