If the set of natural numbers is closed under addition, how can we have the result that the sum of all the natural numbers to infinity is -1/12

Question

As seen here and on this wikipedia page the sum of all the natural numbers to infinity is -1/12.

$\sum_{n=1}^\infty n = \frac{-1}{12}$

but the set of natural numbers is closed under addition and $\frac{-1}{12}$ is not a natural number. In addition the series is clearly divergent, so how can we get away with "assigning" is a value as described on the wikipedia page.

Answer 1

This is true in string theory, which has $26$ dimensions. (Euler proved it) Also, they make assumptions that are not true in "normal" mathematics with the stand axioms.

They assume things like $\displaystyle{\sum_{n=1}^{\infty}} (-1)^n = 1/2$ which clearly is not true under our axioms.

Answer 2

Notice that this closure is closure of a finite number of terms/summands; $1+2+3+4+....+n+...$ is not an integer (nor even a Real number). Notice the same is the case for Rational numbers; $e=e^1=1+1/2+1/3!+....$ where we should use'='; we need the quote, since this is not strict equality; notice that when you do an infinite sum, you do not have strict equality , but instead, you need to deal with issues of convergence instead.