There’s a video doing the rounds that purports to show 1 + 2 + 3 + 4 … = -1/12.

Briefly the sum 1 + 2 + 3 + 4 … does not converge. As a limit we would say the sum approaches positive infinity.

There are some physics and maths applications in which there are other kinds of “sums”, but the video doesn’t give much hint that is dealing with anything other than arithmetic summation so, dealing with it on its own terms, it is wrong, because of two important errors.

Firstly, you can’t perform additions, subtractions etc on “infinities” as though they were ordinary numbers. If you do allow that, you can prove anything.

1 + 2 + 3 + 4 + 5 … = X

Now add 0 to both sides

0 + 1 + 2 + 3 + 4 … = 0 + X

Subtract the second eqn from the first

1 + 1 + 1 + 1 + 1 … = 0

So the sum of an infinite series of 1s is 0, yay.

Add another 0 to both sides …

0 + 1 + 1 + 1 + 1 … = 0 + 0

Now subtract that last eqn from the second last eqn …

1 + 0 + 0 + 0 + 0 + 0 = 0

Therefore, 1 = 0

The second error is that they use a power series expansion outside the domain for which it is valid. The power series representation of 1/(1 + x)^2 is 1 – 2x + 3×2 – 4×3 + · · for a domain of -1 < x < 1.
The domain is important. This formula is invalid outside that domain. That is, it is not valid for x=1, x=-1, or x> 1 or x < -1. Outside that range, the sum diverges.

—-

So, going through the “proof” in the video:

S = 1 + 2 + 3 + 4 + 5 + 6 + …

4S = 4 + 8 + 12 + …

Subtract 4S from S and get

-3S = 1 – 2 + 3 – 4 + 5 – 6 …

This is where the first error is. You can’t subtract infinite numbers like you would finite numbers.

But ignoring that and moving on to the next part

The RHS of that last equation looks like the power series expansion of 1/(1 + x)^2 for x = 1… that is, 1/(1+1)^2

This is the second error. That power series expansion has a domain of -1 < x < 1, so saying it is the power series expansion of 1/(1 + 1)^2 is a nonsense.

So from there, you go

-3S = 1/(1+1)^2 = 1/4
S = -1/12

Thanks for that.

:)

to do it more formally you need Riemann Zeta Function magics

1 – 1 + 1 – 1 … = 1/2
1 – 2 + 3 – 4 … = 1/4
1 – 2 + 4 – 8 … = 1/3
If I remember correctly. PS. I proved these recently, my calculations available on request.
1 + 2 + 3 + 4 … = Triangular numbers = \omega*(\omega+1)/2 where \omega is ordinal infinity.

mollwollfumble said:

1 – 1 + 1 – 1 … = 1/2
1 – 2 + 3 – 4 … = 1/4
1 – 2 + 4 – 8 … = 1/3
If I remember correctly. PS. I proved these recently, my calculations available on request.
1 + 2 + 3 + 4 … = Triangular numbers = \omega*(\omega+1)/2 where \omega is ordinal infinity.

The proof in the video is wrong because it requires “shift invariance”. “Shift invariance” can only be used in evaluating convergent series. My proofs for the other three evaluations do not use shift invariance.

As an example of how shift invariance fails, consider 1 – 1 + 1 – 1 …
If bracketed as (1 – 1) + (1 – 1) + (1 … it sums to 0 + 0 + … = 0
If bracketed as 1 (- 1 + 1) + (- 1 + 1) … it sums to 1 + 0 + 0 + … = 1.
The bracketing is allowed by shift invariance, which gives wrong answers.

This video

https://www.youtube.com/watch?v=w-I6XTVZXww

shows using the Reinmann Zeta function.

sibeen said:

This video

https://www.youtube.com/watch?v=w-I6XTVZXww

shows using the Reinmann Zeta function.

there is also some interesting discussion at the end of the video about the nature of infinite sums and the implications these results have on modern physics… This is, for me at least, the really interesting bit…

more here…

https://www.youtube.com/watch?v=0Oazb7IWzbA&ebc=ANyPxKpoKaiwHFlaskUdftWT2cu7CUFUEMI-dzO9vg23Vsa3UaKkKB7EDUh-M1yWfLu3AiMc_BB3DO1JoquIjLY5VnFiOqw2aw&nohtml5=False

sibeen said:

This video

https://www.youtube.com/watch?v=w-I6XTVZXww

shows using the Reinmann Zeta function.

He produced Ghostbusters as well

OMG… I think I’ve just been consumed by Numberphile videos…

diddly-squat said:

OMG… I think I’ve just been consumed by Numberphile videos…

That was me last week. Some very interesting ones amougst it all.

diddly-squat said:

OMG… I think I’ve just been consumed by Numberphile videos…

poikilotherm said:

diddly-squat said:

OMG… I think I’ve just been consumed by Numberphile videos…

diddly-squat said:

OMG… I think I’ve just been consumed by Numberphile videos…

dv said:

So, going through the “proof” in the video:
S = 1 + 2 + 3 + 4 + 5 + 6 + …

4S = 4 + 8 + 12 + …

Subtract 4S from S and get

-3S = 1 – 2 + 3 – 4 + 5 – 6 …

This is where the first error is. You can’t subtract infinite numbers like you would finite numbers.

But ignoring that and moving on to the next part

The RHS of that last equation looks like the power series expansion of 1/(1 + x)^2 for x = 1… that is, 1/(1+1)^2

This is the second error. That power series expansion has a domain of -1 < x < 1, so saying it is the power series expansion of 1/(1 + 1)^2 is a nonsense.

I should have read the original post in detail first. Yes, that’s exactly correct.

I’m beginning to get annoyed that mathematicians have not understood the difference between “potential infinity” ∞ and “actual infinity = ordinal infinity” ω.

Potential infinity is defined as “∞ is the only number that satisfies ∞ = ∞ +1”.
Clearly, you can’t subtract ∞ from both sides of the equation or you get 0 = 1.

Ordinal infinity can be defined in ten or so equivalent ways, each of which boils down to “ω is the number of natural numbers”.
Here, ω ≠ ω + 1.
If you subtract ω from both sides of the equation you get 0 ≠ 1, which is correct.
ω can be manipulated algebraically with perfect safety, there’s even a branch of mathematics called “nonstandard analysis = hyperreal numbers” that lets you construct a unique factorisation of ω.

diddly-squat said:

sibeen said:

This video

https://www.youtube.com/watch?v=w-I6XTVZXww

shows using the Reimann Zeta function.

there is also some interesting discussion at the end of the video about the nature of infinite sums and the implications these results have on modern physics… This is, for me at least, the really interesting bit…

I had occasion to use the Riemann Zeta in anger recently, I needed to sum a series and the result was a value of the Riemann Zeta function. There’s nothing magical about it, you can get paradoxes by using the Riemann Zeta function wrongly just as easily as by using elementary algebra wrongly. I was shocked at how little has actually been published about exact finite values of the Riemann Zeta function.

Yes. I recently applied my knowledge of infinite sums to modern physics. The application was as follows.

The evaluation of the path integral in lattice QCD, in the absence of Wick rotation, generates an oscillating divergent infinite series and this is known in physics as the “numerical sign problem” or “sign problem”. Expressing the integral in polar coordinates makes the divergence worse, but integration by parts makes this divergence less until the divergence turns into convergence. The subtlety is that doing the integration by parts spins off values like f(ω)cos(ω) that need to be evaluated using a knowledge of the infinite number ω. I do that using cos(ω) = 0.

I should make my own video. How should I go about it?

mollwollfumble said:

dv said:

So, going through the “proof” in the video:
S = 1 + 2 + 3 + 4 + 5 + 6 + …

4S = 4 + 8 + 12 + …

Subtract 4S from S and get

-3S = 1 – 2 + 3 – 4 + 5 – 6 …

This is where the first error is. You can’t subtract infinite numbers like you would finite numbers.

But ignoring that and moving on to the next part

The RHS of that last equation looks like the power series expansion of 1/(1 + x)^2 for x = 1… that is, 1/(1+1)^2

This is the second error. That power series expansion has a domain of -1 < x < 1, so saying it is the power series expansion of 1/(1 + 1)^2 is a nonsense.

I should have read the original post in detail first. Yes, that’s exactly correct.

I’m beginning to get annoyed that mathematicians have not understood the difference between “potential infinity” ∞ and “actual infinity = ordinal infinity” ω.

Potential infinity is defined as “∞ is the only number that satisfies ∞ = ∞ +1”.
Clearly, you can’t subtract ∞ from both sides of the equation or you get 0 = 1.

Ordinal infinity can be defined in ten or so equivalent ways, each of which boils down to “ω is the number of natural numbers”.
Here, ω ≠ ω + 1.
If you subtract ω from both sides of the equation you get 0 ≠ 1, which is correct.
ω can be manipulated algebraically with perfect safety, there’s even a branch of mathematics called “nonstandard analysis = hyperreal numbers” that lets you construct a unique factorisation of ω.

I don’t follow that at all at all.

Surely the number of natural numbers is not a natural number, so the statement ω ≠ ω + 1 does not actually mean anything, since ω + 1 is not defined.

Apparently in the entire history of the electric internet, exactly zero people have asked the exact question:

“is the number of natural numbers a natural number”

mollwollfumble said:

I should make my own video. How should I go about it?

I’d imagine that getting a video camera would be the first step.

sibeen said:

mollwollfumble said:

I should make my own video. How should I go about it?

I’d imagine that getting a video camera would be the first step.

thanks dad

The Rev Dodgson said:

Surely the number of natural numbers is not a natural number, so the statement ω ≠ ω + 1 does not actually mean anything, since ω + 1 is not defined.

There are many definitions for ω + 1.

Cantor used the definition
ω + 1 = {1,2,3,4,…,1} ie. the set of natural numbers with an extra 1 added on the end.

John Horton Conway used the definition of surreal numbers
ω = (N|) ie. the Dedekind cut with the set of natural numbers on the left and the empty set on the right.
ω + 1 = (ω|) ie. the Dedekind cut with the set containing the single number ω on the left and the empty set on the right.

Robinson in his hyperreal numbers defined ω + 1 using what is called Ultrapower construction

The definition of ω + 1 from Hahn series is different again.

I’ve been working with a different definition, loosely based on Cauchy series:
ω + 1 = lim_{i → ω} (i+1)

These definitions are all equivalent, and in each case ω ≠ ω + 1.

Having looked further…

The plans appear to be a little sketchy at present but it does involve using tight laser aimed right at earth, in order to bring down the transmission power requirements. The technology to do this doesn’t exist yet so that’s one of the things on his to-do list.

Wong, Fred.

diddly-squat said:

sibeen said:

mollwollfumble said:

I should make my own video about infinite numbers. How should I go about it?

I’d imagine that getting a video camera would be the first step.

have a video camera.

It would help enormously if there was some not-too-expensive software that would allow me to animate by: moving a textbox around the screen (using keyframes) and animate a simple sketch (eg. Achilles and tortoise, moving along a timeline).

Possible in Powerpoint?
Possible in Java?