# What is 3 2 1 12

## The sum of all natural numbers

One of the most remarkable equations in science is that the sum of all natural numbers - the sum of 1, 2, 3, and so on to infinity - is assigned the value -1/12. This is not a joke and even plays a role in physics. But how can the equation be correct?

The two mathematicians in the worth seeing clip from Brady Haran's Numberphile series demonstrate so vividly how one arrives at this result that one can hardly resist their powers of persuasion. And yet: Quite a few other mathematicians have their hair on end.

So what is it about when you dig a little deeper than the video? The sum of 1, 2, 3 to infinity is a divergent series: With each addend it gets bigger and bigger, and there is never a shortage of further addends.

As soon as such a series appears in a calculation, it prevents mathematicians and physicists from calculating further. In the 18th and 19th centuries, respectively, Leonhard Euler and Bernhard Riemann tackled the problem of divergent series through the trick of "analytical continuation". They cleverly defined new series that are convergent for some values โโ- that is, despite an infinite number of summands, they do not become infinitely large, but rather reach a defined limit value. For other values, on the other hand, their terms are just the terms of the original divergent sum.

One of these new series is defined as the sum 1 + 1/2^{s} + 1 / 3^{s} + ..., where s is a so-called complex number. For certain values โโof s it is identical to the (otherwise remarkable) Riemann zeta function (this is how the zeta function is even defined). The highlight: This function also has a clearly defined value there - namely at s = -1 - where the associated series is identical to the sum of 1, 2, 3 ... Namely the value -1/12.

The result continues to contradict our common sense. But we don't need to be impressed by this: Even mathematicians are worried about divergent series - infinite numbers are just hard to grasp.

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