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[–] 21 points22 points  (0 children)

"somewhat unintuitive analytic continuation"

...boy that's an understatement

[–]Number Theory 7 points8 points  (0 children)

This is a good post also in the topic:

http://davidlowryduda.com/?p=1551

[–] 9 points10 points  (3 children)

As Tao points out, the manipulations used to produce the -1/12 result (on Reddit and Numberphile, not by professional mathematicians) are usually not justified and can lead to contradictions.

[–] 0 points1 point  (2 children)

Line by line, I think I understand most of what Tao was doing. However, it seems to me that if those tricks lead to contradictions, algebra in general is to blame.

The Numberphile proof is built on this proof:

x = 1-1+1-1+...

x = 1 - x

2x = 1

x = 1/2

Can you tell me which step is invalid, and why?

[–] 3 points4 points  (0 children)

Yes, the first step is invalid.

Think about the domain of your operations. You are presumably working in the field of real or complex numbers. Therefore, any variable you declare has to take on a value inside one of those fields. Just as you cannot indiscriminately declare a variable x to be infinity, you cannot declare x to be (1-1+1-1+...) unless the latter has a clear-cut meaning as a member of the algebraic structure you are working in.

As an ordinary (Cauchy) sum, (1 - 1 + ...) does not exist. However, if you change the interpretation of the sum, then things actually can and do change. If we interpret 1 - 1 + ... as a Cesaro-sum, it turns out the sum exists, so the assignment makes sense. Moreover, by some elementary properties of that sum, it can be shown that x = 1 - x in fact is true. In this case the proof does go through, with the proviso that we are talking about the Cesaro-sum of the series.

[–] 1 point2 points  (0 children)

The step x = 1 - x isn't valid as written, since there's no rigorous interpretation of what "x" means in this case (it's divergent in the usual sense, but also Cesaro summable; so x can be undefined or have a finite value, depending on the method of summation).

[–]Math Education 3 points4 points  (1 child)

I took a class on problem solving in my undergrad, which was basically a series of trick questions. Some of them were like " Benny does so and so to prove 1 =0. Is he right?"

In that class there was a problem similar to this, where we ultimately decided that the logic was flawed because the guy treated divergent sums like algebraic variables, and you can't do that. I also don't accept that 1-1+1-1.... Converges to 1/2. I'll check out the other proofs and see if they change my mind about that though.

[–] 10 points11 points  (0 children)

I also don't accept that 1-1+1-1.... Converges to 1/2

The clue here is what do you mean by 'converge'. Convergence is dependent on the topology you endow the space with, and the summation method used. The series clearly does not converge in the usual sense, but it is Cesaro-summable and Abel-summable to 1/2.
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