Why is the sum of all natural numbers equal to -1/12?


When we add numbers, increasing each next term by the same positive value, then we will get more and more values ​​with each new term. For example, the sum of all natural numbers from 1 to 10 is 55, from 1 to 100 will be 5050, and due to an increase in the upper bound, the sum will become larger. It is logical to assume that the sum of numbers from 1 to infinity will give an infinitely large number, but if we do the calculations, we get a value of -1/12. It’s one thing if we got a certain finite number with a plus sign, although this causes dissonance, it’s another matter when we get a fraction with a minus sign, which looks completely absurd, because how can we get negative when adding positive numbers. In this article, we will take a little look at how mathematicians generally got such a meaning.

To do this, first it is worth analyzing what some of the other amounts look like. Let’s start with this: 1-1 + 1-1 + 1-1 … .. This is an infinite sum of ones and minus ones, which alternate. If we knew what sign the given numerical series ends with, then we would have no problem saying what its final value is, but this series is infinite. Let’s designate this set of numbers as A, for convenience. In order to calculate the amount, after the first element we put out the minus, as shown in the figure below. Since the series of numbers is infinite, the expression in brackets will also be the sum of A and after ordinary algebraic calculations we get that A = 1/2, more specifically, then 1-1 + 1-1 + 1-1 + 1 … = 1/2

Now let’s count the next row: 1-2 + 3-4 + 5-6 + 7-8 + … Let’s call this row number B and now add to it the already counted row A = ½, but add starting from the second element like this, as shown in the diagram below. Then we get that the same series B is subtracted from unity. After the usual algebraic actions, we get B = ¼

And finally, that same series is the sum of all natural numbers, let us denote it by X. X = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + … Subtract from it the above-mentioned series of numbers B = ¼. As a result, we get: 1-1 + 2 + 2 + 3-3 + 4 + 5-5 + 6 + 6 + 7-7 …, and for simplicity, we will rewrite it as in the picture below. As a result, it turns out that our sum of all natural numbers X will be subtracted ¼ will be equal to 4 + 8 + 12 + 16 … The sum of all numbers will be a multiple of four, from which we just put 4 out of the parenthesis, and in parentheses we get 1 + 2 + 3 + 4 …, that is, the same row of X, well, after the simplifications shown below, we get X = -1 / 12. Since X = 1 + 2 + 3 + 4 + 5 + 6 …, the sum of all natural numbers is -1/12.

In all this, it is noteworthy that this fact about the sum of numbers is used in many physics articles, especially in quantum mechanics, for example, many articles on string theory operate with it, and also thanks to it, the Casimir Effect was explained. Of course, it is possible and worth questioning the fact that the sum of all natural numbers is equal to a negative number, and perhaps this is a lack of methods for counting series, but the fact that the number -1/12 is used in many scientific articles, and it works, says a lot. …

Author: Vladislav Kigim. Edited by Fedor Karasenko.

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