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.

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 !

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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_______________________________ It's only a tatter of mime

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."

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