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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.

Matthew Kemnetz
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    It is closed under finite addition; $1+1+1.....$ is not a natural number. – user99680 Feb 24 '14 at 21:17
  • But the series is still divergent nonetheless. – Matthew Kemnetz Feb 24 '14 at 21:20
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    Maybe you would be more content with $\sum_{n=1}^ \infty = \infty$? Well, $\infty$ is not a natural number either! How about something that has a "traditional" limit: The rationals are closed under additoin, too. But $\sum_{n=1}^\infty \frac1{n^2}=\frac{\pi^2}6$ is irrational. You really have to distinguis sums from series! – Hagen von Eitzen Feb 24 '14 at 21:20
  • The link to Wikipedia does not work. What do you mean by "getting away with 'assigning' a value"? – JiK Feb 24 '14 at 21:21
  • From the wiki: "Many summation methods are used in mathematics to assign numerical values even to divergent series. In particular, the methods of zeta function regularization and Ramanujan summation assign the series a value of −1/12, which is expressed by a famous formula"

    I am sorry but I can't get the link to work here

     – Matthew Kemnetz Feb 24 '14 at 21:25
  • This result is counterintuitive, which would give the average mathematician pause. I used to simply assume that such a sum was incorrect. And I'm sure our brothers in number theory and calculus would probably take issue with this result. As the term of the sum approaches infinity, the sum should approach infinity, no matter what string theory predicts. – Brian J. Fink Feb 24 '14 at 21:37

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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.

user99680
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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.

Skuttle_Butt
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  • That is the assumption upon which the entire sum hangs, but I would argue that it is utter nonsense to simply average the odd and even result. The problem I see is that it would have to have a value, but we don't know that value because, well, is infinity even? How could we measure it to find out? – Brian J. Fink Feb 24 '14 at 21:47