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Yeah, but infinity isn't a number or a thing that actually exists, it's a thought experiment. In a world where infinity exists, nothing is as it seems and lots of rules get broken.
And as they say in the article the sum is only -1/12 because they redefine what " = " means.
Particularly the first step where they define S1 as this series
S1 = 1 - 1 + 1 - 1 + 1 - ...
and then say this series has the value 1 if you stop it at an odd point (correct) and that it has the value 0 if you stop it at an even point (also correct)
...but then they handwave a bit and define this series to have the value 1/2 ("because we don't know where we'll stop so we'll take the average")...but this is not a valid way of doing math because this series
never ever converges to 1/2 no matter where you stop ... writing
S1 = 1/2 is therefore not a valid equality.
They specifically state that this is used in
physics (not math!)...and there this is absolutely OK because in physics no such thing as a true infinity exists and you can get good results with approximations. But as a mathematical proof it just doesn't work that way. "Approximately good enough" is not an valid operation in a mathematical proof.
What you could do is split the proof into two separate proofs using in the first one the series
S1_0 = 1 - 1 + 1 - 1 + 1 - ... broken off at an odd number such that S1_0 = 1
and then try to show the result is -1/12
and then make a second proof using the series
S1_1 = 1 - 1 + 1 - 1 + 1 - ... broken off at an even number such that S1_1 = 0
and then try to show the result is -1/12
But you will quickly see that neither works.
Last edited by antialias; 08-07-2018 at 10:53 AM.
