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The following infinite sums produce remarkable results.

$1+2+3+4+...=-\frac{1}{12}$

$1-2+3-4 +...=\frac{1}{4}$

So how are these results compatible with the statement; that integers are closed under addition? Analysis and algebra seem to disagree here.

Is this some sort of abuse of notation? Is this an example of inconsistent statements under the logicians meaning of the word inconsistent?

And are there any other examples of seemingly contradictory statements from history or presently?

Note. I'm not asking for a proof of the results.

Brad Graham
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  • https://www.google.de/search?q=1%2B2%2B3%2B4&ie=utf-8&oe=utf-8&gws_rd=cr&ei=jNnRVtGxO4vjywOUqY_4Aw#q=1%2B2%2B3%2B4%2B....+stack+exchange

    first link. search the site properly!

     – tired Feb 27 '16 at 17:15
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    By definition, “closed” under addition means closed under finite sums. There is no inconsistency, even with regard to similar, but convergent series like $\sum_{k=1}^\infty \dfrac{1}{k^2}$. The rationals are closed under addition, but this sum is irrational. – Steve Kass Feb 27 '16 at 17:23
  • "are there any other examples of seemingly contradictory statements?" You may as well be asking for a list of abuses of notation or confluences of terminology, which I think you can probably find online. – Mark S. Feb 27 '16 at 17:25
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    I don't understand, mathematics has a vast array of interesting stuff and this $-{1 \over 12}$ things gets so much attention. If you add positive numbers you end up with a positive number. The rest is trickery, like 'proof's of $1=0$. I suppose its entertainment. Hmm, is Trump involved in this? – copper.hat Feb 27 '16 at 17:36
  • @ copper Perhaps its more accessible, as a realisation to study studying itself, than some of the interesting stuff you feel is neglected. – Brad Graham Feb 27 '16 at 17:43

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Addition, the operation that integers are closed under, is a binary operation: taking two inputs, so $ a+b $ only. By an induction argument, closure under addition implies closure under finite sums since finite sums are defined in terms of repeated binary sums.

However, none of the ways to define infinite sums are just "repeated application of binary addition" so your examples just show something like "for some of the nonstandard methods for defining the 'sum' of an infinite sequence, the integers aren't closed under that operation".

The fact that $+$ signs are used is potentially confusing here since binary addition is not being used. That's the abuse of notation.

Mark S.
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  • So is there an algebraic object that is defined by infinitary operations? – Brad Graham Feb 27 '16 at 17:33
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    @BradGraham, most often, "algebra" is related to finitary operations (look up "universal algebra", for instance). There are lots of infinitary operations like those described on the Wikipedia page for "divergent series" or "linear functionals on $\ell_2$" or even just the supremum of a bounded sequence of real numbers. – Mark S. Feb 27 '16 at 19:27