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PostPosted: Mon Aug 13, 2018 3:42 pm 
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DISCLAMER: Equations below are not exactly right. But they are not exactly wrong either.

Here's a fun challenge: explain at which steps what properties of infinity have been abused without any advanced math jargon.
No "methods of summation", no "convergence/divergence", no "limit" or "approaches sth", or even "function". Turning the usual 'proof' backwards helps intuition A LOT.
You can do it !

Negative numbers are highlighted blue, because 1) the minuses are way too small and unnoticeable, 2) it helps to notice some patterns, and 3) it looks pretty.

-1/12 =

-1/12 + 0 + 0 + 0 + 0 + 0 + … =

-1/12 + 0/12 + 0/12 + 0/12 + 0/12 + 0/12 + … =

-1/12 + (1-1)/12 + (1-1)/12 + (1-1)/12 + (1-1)/12 + (1-1)/12 + … =

-1/12 + (-1/12 + 1/12) + (1/12 + -1/12) + (-1/12 + 1/12) + (1/12 + -1/12) + (-1/12 + 1/12) + … =

(-1/12 + -1/12) + (1/12 + 1/12) + (-1/12 + -1/12) + (1/12 + 1/12) + (-1/12 + -1/12) + (1/12 + 1/12) + … =

-2/12 + 2/12 + -2/12 + 2/12 + -2/12 + 2/12 + … =

-1/6 + 1/6 + -1/6 + 1/6 + -1/6 + 1/6 + ... =

(-1+0)/6 + (1+0)/6 + (-1+0)/6 + (1+0)/6 + (-1+0)/6 + (1+0)/6 + ... =

(-1+(0-0))/6 + (1+(1-1))/6 + (-1+(2-2))/6 + (1+(3-3))/6 + (-1+(4-4))/6 + (1+(5-5))/6 + ... =

(0-1)/6 + (2-1)/6 + (2-3))/6 + (4-3)/6 + (4-5)/6 + (6-5)/6 + ... =

-1/6 + (-1/6 + 2/6) + (2/6 + -3/6) + (-3/6 + 4/6) + (4/6 + -5/6) + (-5/6 + 6/6) + ... =

(-1/6 + -1/6) + (2/6 + 2/6) + (-3/6 + -3/6) + (4/6 + 4/6) + (-5/6 + -5/6) + (6/6 +6/6) + ... =

-2/6 + 4/6 + -6/6 + 8/6 + -10/6 + 12/6 + ... =

-1/3 + 2/3 + -3/3 + 4/3 + -5/3 + 6/3 + ... =

-1/3 + (2+0)/3 + -3/3 + (4+0)/3 + -5/3 + (6+0)/3 + ... =

-1/3 + (2+(2-2))/3 + -3/3 + (4+(4-4))/3 + -5/3 + (6+(6-6))/3 + ... =

-1/3 + (4-2)/3 + -3/3 + (8-4)/3 + -5/3 + (12-6)/3 + ... =

-1/3 + (4/3 + -2/3) + -3/3 + (8/3 + -4/3) + -5/3 + (12/3 + -6/3) + ... =

(-1/3 + 4/3) + (-2/3 + 8/3) + (-3/3 + 12/3) + (-4/3 + 16/3) + (-5/3 + 20/3) + (-6/3 + 24/3) + ... =

3/3 + 6/3 + 9/3 + 12/3 + 15/3 + 18/3 + … =

1 + 2 + 3 + 4 + 5 + 6 …

To summarize,
-1/12 = 1 + 2 + 3 + 4 + 5 + 6 + ...

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PostPosted: Mon Aug 13, 2018 9:23 pm 
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Not even bothering with the steps beyond this bit:
(-1/12 + -1/12) + (1/12 + 1/12) + (-1/12 + -1/12) + (1/12 + 1/12) + (-1/12 + -1/12) + (1/12 + 1/12) + … =
Because that is not equivalent to the previous sequences. The end term pulls 1/12 from the next implied (-1/12 + 1/12) of the previous sequence without showing the -1/12 that is part of the zero equivalency.

Glancing at the conclusion, it's like saying 1 = infinity, as long as all the negative numbers are swept behind the ellipses.

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PostPosted: Tue Aug 14, 2018 12:25 am 
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Technically, you are right - those two expressions are not really equal.
However, think about this:
If we stop on any even term, we "borrow" +1/12 from the unaccounted future steps.
If we stop on any odd term, we "borrow" -1/12 from the unaccounted future steps.
However, an infinite number of repetitions is neither even nor odd. (Which is a property of infinity we exploit on this step.)
So, because the two cases are equally distributed and do not change, we can* average between them.

Also, the conclusion is more like "-1/12 = 1 + 2 +3 + ..., as long as we care about free redistribution of numbers in series more than about their partial sums."


*Except not really.

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PostPosted: Mon Aug 20, 2018 8:45 pm 
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The sum of infinite natural numbers relies on a massive fallacy as far as I'm concerned; basically it assumes that the last term (infinity for all intents and purposes) is both even and odd, which unfortunately cannot be the case as a number is exclusively odd, even, or neither in the case of irrational numbers, imaginary numbers, and otherwise non-integers. Even if you take the average of two sequences, one assuming infinity is odd and the other infinity is even, then you have the average of an odd and an even, which always results in a non-integer rational number, and thus throws the whole pattern out of the window.


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PostPosted: Tue Aug 21, 2018 1:10 am 
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The sum of infinite natural numbers relies on a massive fallacy as far as I'm concerned; basically it assumes that the last term (infinity for all intents and purposes) is both even and odd, which unfortunately cannot be the case as a number is exclusively odd, even, or neither in the case of irrational numbers, imaginary numbers, and otherwise non-integers. Even if you take the average of two sequences, one assuming infinity is odd and the other infinity is even, then you have the average of an odd and an even, which always results in a non-integer rational number, and thus throws the whole pattern out of the window.

you know what's a non-integer rational number, though? -1/12.

:duel:

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