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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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