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Greyman wrote:Nice try, but the very first premise is blatantly erroneous leading to a mathematical fallacy.
The sum is indefinite because the series does not converge. You do not just take the average of the partial sums.
It's the equivalent of hiding a divide by zero step to prove 2=1.
1. Let a and b be equal non-zero quantities
a = b
2. Multiply through by a
a^2 = ab
3. Subtract b^2
a^2 - b^2 = ab - b^2
4. Factor both sides
(a - b)(a + b) = b(a - b)
5. Divide out (a - b)
a + b = b
6. Observing that a = b
b + b = b
7. Combine like terms on the left
2b = b
8. Divide by the non-zero b
2 = 1
QED
Paradoxica wrote: You cannot approach it with standard mathematics. It requires far more advanced mathematics than the standard human is expected to know.
Sendraks wrote:Non-standard humans? What are we talking about here?
Paradoxica wrote:Mathematicians and Quantum Physicists.
VazScep wrote:Smells like bullshit to me.
I'll agree that ζ(-1) = -1/12, but I can't see anyone giving a purely mathematical reason why this should be taken as a sensible value for 1 + 2 + 3 + ....
There's plenty of sense to be found in the standard treatment of infinite series, where you just say it's the number you eventually get as close as you like to by adding more and more of the summands.CdesignProponentsist wrote:I would smell bullshit if there was a sensible answer. Dealing with infinite series rarely ever results in sensible answers.
VazScep wrote:S = -1/2
D'oh. Yeah.
There are principled reasons for allowing non-Euclidean geometries and permitting a square root of -1.And no, this is not bullshit at all. Calling it bullshit is like saying that the square root of -1 is bullshit, or that a triangle for which the sum of the angles isn't 180° is bullshit. Sure, for those who've never heard of complex numbers and non-Euclidean geometry, it looks like nonsense. But that's only because they haven't learned the more advanced stuff.
All that matters? No. Mathematics isn't just derivations from arbitrary sets of consistent axiom systems and random definitions.All mathematical statements are derived from some set of axioms. These axioms can be changed, from which new statements can be derived. All that matters is that the results are logically consistent.
So this is getting to some sort of justification, though as I said above, I'd like to see some purely mathematical justifications, rather than one's motivated by physics.In this case, one can change the traditional criterion of convergence, and derive these results. In particular, it has to do with Zeta function regularization.
VazScep wrote:There's plenty of sense to be found in the standard treatment of infinite series, where you just say it's the number you eventually get as close as you like to by adding more and more of the summands.CdesignProponentsist wrote:I would smell bullshit if there was a sensible answer. Dealing with infinite series rarely ever results in sensible answers.
Sometimes, there is no such number for a given series, but you can come up with silly alternatives if you pretend it exists and then start doing algebra. For instance:
S = 1 + 2 + 4 + 8 + 16 + ...
2S = 2 + 4 + 8 + 16 + ...
2S + 1 = 1 + 2 + 4 + 8 + 16 + ... = S
S = -1/2.
However, this is just bullshit.
The videos above look like a combination of bullshit algebra and then a bullshit claim that values of the zeta function are generally sums of an infinite series. There might be some reason from physics to get away with this with a wink and a grin, but I can't see any serious mathematical reason to do this.
It isn't. You need definitions in place to prove something like this, and you need to motivate those definitions. All of that is absent from the video.CdesignProponentsist wrote:I'm not a mathematician but from what they said in the video, this isn't just mathematical trickery, but what they provided was an actual proof.
Ramanujan summation essentially is a property of the partial sums, rather than a property of the entire sum, as that doesn't exist.
"Dear Sir, I am very much gratified on perusing your letter of the 8th February 1913. I was expecting a reply from you similar to the one which a Mathematics Professor at London wrote asking me to study carefully Bromwich's Infinite Series and not fall into the pitfalls of divergent series. … I told him that the sum of an infinite number of terms of the series: 1 + 2 + 3 + 4 + · · · = −1/12 under my theory. If I tell you this you will at once point out to me the lunatic asylum as my goal. I dilate on this simply to convince you that you will not be able to follow my methods of proof if I indicate the lines on which I proceed in a single letter. …"[
VazScep wrote:There are principled reasons for allowing non-Euclidean geometries and permitting a square root of -1.
VazScep wrote:Mathematics isn't just derivations from arbitrary sets of consistent axiom systems and random definitions.
Thommo wrote:I watched most of the first clip and stopped, what he calls a proof isn't a proof, he performs operations without defining them or showing they have the properties he relies on (e.g. providing the definition for taking the arithmetic mean of the two values as the sum of S1 or showing that placewise addition of terms displaced by one position as used to derive S2 represents addition of sequences - i.e is commutative, associative, has an identity and inverses) which are nonstandard for the notation he uses. It may well be that they have the properties and the result is valid (although confusing as the notation matches that for divergent sequences and summation via epsilon definitions) but there's no possible way to conclude that from the video. Presumably the idea is to inspire people to go and learn some new mathematics, but personally I'm turned off by the claim of proof and bypassing of the treatment via Riemann zeta function which would make a lot more sense.
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