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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.

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    $\begingroup$ It is closed under finite addition; $1+1+1.....$ is not a natural number. $\endgroup$
    – user99680
    Feb 24, 2014 at 21:17
  • $\begingroup$ But the series is still divergent nonetheless. $\endgroup$ Feb 24, 2014 at 21:20
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    $\begingroup$ 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! $\endgroup$ Feb 24, 2014 at 21:20
  • $\begingroup$ The link to Wikipedia does not work. What do you mean by "getting away with 'assigning' a value"? $\endgroup$
    – JiK
    Feb 24, 2014 at 21:21
  • $\begingroup$ 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 $\endgroup$ Feb 24, 2014 at 21:25

2 Answers 2

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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.

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  • $\begingroup$ That's a good observation (about "summing" infinitely many rationals). +1 for you. $\endgroup$
    – MPW
    Feb 24, 2014 at 22:09
  • $\begingroup$ @MPW: I appreciate it, thanks. $\endgroup$
    – user99680
    Feb 24, 2014 at 22:10
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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.

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  • $\begingroup$ 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? $\endgroup$ Feb 24, 2014 at 21:47

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