1 + 2 + 3 + ... = -1/12

[AnnoyedOne]

(10-26-2024 06:02 PM)Albert Chan Wrote:  10 years later, The Return of -1/12 - Numberphile

Quote:"Oh, that was easy," says Man, and for an encore goes on to prove that black is white and gets himself killed on the next zebra crossing.

― Douglas Adams, The Hitchhiker’s Guide to the Galaxy

I've long believed that many mathematicians and physicists "went off the rails" some time ago. Reality is just a plaything to them. Engineers on the other hand don't want to "get killed".

FYI I would challenge the claim that there's any such thing as infinite anything. So 0.999.... might be "close enough" to "1" but it isn't "1".

A1

[Thomas Klemm]

(10-26-2024 06:27 PM)AnnoyedOne Wrote:  FYI I would challenge the claim that there's any such thing as infinite anything. So 0.999.... might be "close enough" to "1" but it isn't "1".

For a proof of your claim just grab your favorite HP calculator and enter:

9
ENTER
1/X
×

0.999999999999

Even Free42 gives:

9.999999999999999999999999999999999e-1

And that can't be wrong.
It's true, it is displayed as:

1

But you certainly know how to SHOW its true value.

[Johnh]

Infinite series' are embedded in very many mathematical expressions, that calculate values that we need in the real world, such as trig functions, Pi etc. So for that reason at least, the concept of infinity can enter the real world, and so help us understand it, (even if we needn't actually go to infinity, nor beyond!)

I also accept the point made in the last video above, that if there is a clear convergence to a value, then we can take that value as the valid sum of the infinite series, even though we can't write down a full.infinite series, eg, 0.99, 0.999, 0.9999... etc, converges to 1.

But, I can't accept the conjecture that the the series 1 -1 +1 -1 +1... etc is in some way equal to 0.5. While a converging series can be seen to converge on a value, this series is never either exactly 1/2 nor anywhere near it and that's true no matter how many terms we look at! It is simply an undefined result that may have one of two values after n terms.

So if that argument is key to the idea about -1/12, then I'm missing it!

Here's a practical analogy. A 240V mains voltage is measured over a very long time.. tending to Infinite time. What is the voltage? Average is zero volts! But that doesn't characterise it very well, nor how much it hurts. The added dimension of time is needed to express it properly and usefully. It's maybe a better illustration of the 1 -1 +1... series, if it is a perfect square wave instead of sine, which is never at the average 0V

Do the theoretical physics ideas that use some of these concepts also involve additional dimensions etc?

[Albert Chan]

(10-26-2024 11:08 PM)Johnh Wrote:  I can't accept the conjecture that the the series 1 -1 +1 -1 +1... etc is in some way equal to 0.5.

We are not asking for the series to EQUAL to 1/2. We just ASSIGN 1/2 to the series.

1 + r + r^2 + r^3 + ... = 1/(1-r)

At r = -1, we have:

1 - 1 + 1 - 1 + ... → 1/(1-(-1)) = 1/2





Based from Zeta analytic continuation, we can ASSIGN divergent sum the same way

Zeta(s) → 2^s * pi^(s-1) * sin(pi*s/2) * Gamma(1-s) * Zeta(1-s)

Zeta(-1) → (1/2) * (1/pi^2) * (-1) * 1! * (pi^2/6) = -1/12

[Thomas Klemm]

(10-26-2024 11:08 PM)Johnh Wrote:  But, I can't accept the conjecture that the the series 1 -1 +1 -1 +1... etc is in some way equal to 0.5. While a converging series can be seen to converge on a value, this series is never either exactly 1/2 nor anywhere near it and that's true no matter how many terms we look at! It is simply an undefined result that may have one of two values after n terms.

From Cesàro summation:

Quote:The Cesàro sum is defined as the limit, as n tends to infinity, of the sequence of arithmetic means of the first n partial sums of the series.

The two examples of this article show that the Cesàro sum of the series \(1-1+1-1+1-\cdots\) is 1/2 while \(1+2+3+4+\cdots\) is not Cesàro summable.

[dm319]

(10-03-2018 11:13 AM)Albert Chan Wrote:  Here is a good explanation from Professor Frenkel:

What a great video and explanation. He asks in the video whether the square root of -1 exists. I think the same can be said for negative numbers. I can't really have -3 oranges, but the concept is useful if I owe oranges, and the negative number will become a real number in the future (when I buy 12 oranges in the shop).

[Namir]

It took me a nano-second o look at that formula and say "Nahhhh iit's wrong" and moved on. My time in life is limited and don't wish to waster it away with obvious falacies. Plan to spend my time on new interesting algorithms!

Namir

[Thomas Klemm]

(10-27-2024 12:04 PM)Namir Wrote:  My time in life is limited and don't wish to waster it away with obvious fallacies.

And please don't start talking again about imaginary numbers:

[AnnoyedOne]

(10-26-2024 09:05 PM)Thomas Klemm Wrote:  But you certainly know how to SHOW its true value.

Yes. Plus we could get into the whole issue of "guard" (aka hidden) digits used to make calculators appear more "accurate" than they really are. Round those digits (which you may not be able to see) and you get a "perfect" result. Of course many mathematicians are going to ignore all that and say that f-1(f(x)) = x so it's all good.

In case it wasn't clear the whole premise of the 0.999... = 1 "proof" resets on the concept of infinity (in this case 0.999... recurring to infinity). No such thing IMO except in the mind of some mathematicians/physicists. It's a "Jedi mind trick". This is not the 0.999... you're looking for!

Some of us don't confuse mathematics and/or models/methods with reality. The latter is the final arbiter not math.

My personal favourite is that the mathematical average of a sinewave is zero. Really? Is AC voltage sinusoidal? And the average is zero? Hold these two wires while I crank this alternator! Zero sure feels like a lot huh? Have you ever heard of RMS?

And as Johnh said it's going to hurt whether 110V/120V/220V/240V/250V 50/60Hz. "Megahurts" in fact. Assuming that you live.

In my first year studying engineering there was a teacher whose hands shook pretty badly. Many assumed that he had Parkinsons disease since he was older. One day someone asked. Turns out he'd accidently touched a high voltage line and some nerves had been fried. He showed us the scars.

Which leads me to a few "pet peeves".

"The mass of a photon is zero."
Er, no it isn't. It may be really really small, even unmeasurable, but it has some mass.

"At the center of a black hole is a singularity".
A singularity is a point of zero size and infinite mass. The math (x/0) may predict such a thing but IMO it is likely wrong and/or misinterpreted. Whatever the case you probably don't want to get anywhere near said black hole. That would ruin your day.

"1 + 2 + 3 +... = -1/12"
Really? Is that so? And you can "prove" it? Actually I'm pretty sure that you can't. Your education says that you can? I'd ask for my money back if I were you [blatantly stolen from the 2000 movie Space Cowboys].

A1

[Nigel (UK)]

(10-27-2024 12:36 PM)AnnoyedOne Wrote:  In case it wasn't clear the whole premise of the 0.999... = 1 "proof" resets on the concept of infinity (in this case 0.999... recurring to infinity). No such thing IMO except in the mind of some mathematicians/physicists. It's a "Jedi mind trick". This is not the 0.999... you're looking for!

It isn't hard to prove that \(1 - 0.999\ldots\) is smaller than any positive real number. If you wish to say that it still isn't zero that's absolutely fine; I believe that you can have a completely consistent system of mathematics using such infinitesimal numbers. Equally you can have a completely consistent system of mathematics without them, in which case the difference is zero. You get to choose!

Also, I suggest that even a number like "7" only exists in minds, not in external reality. I've never seen "7" - I've often seen 7 things, and the symbol that represents the number 7 on paper, but I've never encountered the number (or any other number) directly in reality.

Quote:Some of us don't confuse mathematics and/or models/methods with reality. The latter is the final arbiter not math.

The latter is the final arbiter of whether a prediction made by a physical model is correct. However, the fact that (for example) the geometry of space isn't Euclidean (if you look really really closely!) doesn't mean that Euclidean geometry is inconsistent, merely that it doesn't apply universally to reality. Mathematical systems don't have to agree with reality; they only have to be internally consistent.

Quote:Which leads me to a few "pet peeves".

"The mass of a photon is zero."
Er, no it isn't. It may be really really small, even unmeasurable, but it has some mass.

Why? If you mean that "we can never know whether the [rest] mass of a photon is exactly zero" then that's fair enough. However, the zero rest mass is a consequence of a symmetry of Maxwell's equations (gauge invariance) which persists in the quantum version of the theory. If that symmetry were broken then the inverse square law would fail at large distances, charge might not be conserved, and the theory wouldn't obviously be quantizable (although that would be a problem for physicists and not in itself a logical objection). I don't think you can insist that this symmetry cannot be exact.

Quote:"At the center of a black hole is a singularity".
A singularity is a point of zero size and infinite mass. The math (x/0) may predict such a thing but IMO it is likely wrong and/or misinterpreted. Whatever the case you probably don't want to get anywhere near said black hole. That would ruin your day.

No problems here (except for the infinite mass part - infinite density?). People assume that the singularities will go away when there is a proper quantum theory of gravity. Maybe they are right! Who knows?

Quote:"1 + 2 + 3 +... = -1/12"
Really? Is that so? And you can "prove" it? Actually I'm pretty sure that you can't. Your education says that you can? I'd ask for my money back if I were you [blatantly stolen from the 2000 movie Space Cowboys].

Again agree. The "=" sign is being misused here, as it is with \(1-1+1-1+1-1+\ldots=\frac12\). It's all good fun, though!

Nigel (UK)

[AnnoyedOne]

(10-27-2024 04:17 PM)Nigel (UK) Wrote:  Also, I suggest that even a number like "7" only exists in minds, not in external reality.

True. '7' is a concept in the human mind to serve as shorthand for 1+1+1+1+1+1+1 objects (whatever they may be) in reality. It's a relationship. "Understanding" if you wish. However there is no 0.999... recurring to infinity. If I'm wrong show it to me.

But we're getting way off topic here into Metaphysics and Epistemology (Philosophy).

BTW I've never been a fan of Quantum Mechanics either but that is also way OT.

No doubt someone will bring up "smart people". True but throughout history they've made mistakes (or outright deception in some cases) too.

A1

[floppy]

(10-01-2018 11:34 PM)Albert Chan Wrote:  I "know" the sum does not add up to -1/12, but the guys were so serious ...

The problem is people end by thinking there is a flat earth..

[AnnoyedOne]

(10-27-2024 04:17 PM)Nigel (UK) Wrote:  Mathematical systems don't have to agree with reality; they only have to be internally consistent.

I disagree and will leave it at that. Does the expression "castles in the sky" mean anything to you?

As floppy implied some actually believe "nonsense" simply because someone smart claims it.

I've known people way smarter than I. However they could injure themselves badly with a screwdriver so...

A1

[Albert Chan]

(10-27-2024 04:35 PM)AnnoyedOne Wrote:  However there is no 0.999... recurring to infinity. If I'm wrong show it to me.

If x has divisors other than 2 or 5, in decimal, 1/x have infinite digits
We have infinite examples!

1/3 = 3/9 = 0.(3)
3/3 = 0.(3*3) = 0.(9)

1/7 = 142857/999999 = 0.(142857)
7/7 = 0.(142857*7) = 0.(999999) = 0.(9)

1/11 = 09/99 = 0.(09)
11/11 = 0.(09*11) = 0.(99) = 0.(9)
...

We can do above in reverse, getting fraction from repeating decimal

--> 0.(9) = 9/9 = 1

If you are uncomfortable with infinite 9's, we can change it to 0's
Do you agree 1 = 1.000 ... (infinite 0's) ?

If YES, then 1 = 0.999 ... (infinite 9's) must be true, because 1 / 0.999... = 1.000...
Try long division if you are not convinced.

[Albert Chan]

(10-27-2024 12:36 PM)AnnoyedOne Wrote:  "1 + 2 + 3 +... = -1/12"
Really? Is that so? And you can "prove" it? Actually I'm pretty sure that you can't.

We just ASSIGN a finite value to this divergent sum.
Weird sum will get cancelled at the end of calculations.

We could even leave it un-evaluated as zeta(-1), but a number as place-holder is simpler.

F(n) = sum(k, k=n, inf) = zeta(-1) - sum(k, k=1..n-1) = zeta(-1) - n*(n-1)/2

sum(k, k=10..25) = F(10) - F(26) = (zeta(-1)-10*9/2) - (zeta(-1)-26*25/2) = 280

(08-19-2019 04:15 PM)Albert Chan Wrote:  Let \(t = \frac{a}{1-a}\), then \(T= \sum_{x=1}^{\infty} a^x u_x = t \{1 + (tΔ) + (tΔ)^2 + \cdots\}u_1 \)

We assign t = Σ(a^x, x=1..inf) = a/(1-a), even if |a|>1

Below example had a=2 --> t = 2/(1-2) = -2, which make no sense.
But, infinite placeholder will get cancelled, when we ask for finite sum. (*)

(08-19-2019 04:15 PM)Albert Chan Wrote:  

Code:

x  x^2 Forward Difference Table
10 100
11 121 21
12 144 23 2

26 676
27 729 53
28 784 55 2

\(\sum _{10}^{\infty} 2^x x^2 = 2^9 \sum _{1}^{\infty} 2^x (x+9)^2 → 2^9 (100t + 21t^2 +2 t^3) → -67584\)

\(\sum _{26}^{\infty} 2^x x^2 = 2^{25} \sum _{1}^{\infty} 2^x (x+25)^2 → 2^{25} (676t + 53t^2 +2 t^3) → -38788923392\)

\(\sum _{10}^{25} 2^x x^2 = \sum _{10}^{\infty} 2^x x^2 - \sum _{26}^{\infty} 2^x x^2 → 38788855808\)

Amazingly, it match true result

(*) 'infinite' formula was derived from finite sum, but a/(1-a) keep popping out.
It just happened that t = a/(1-a) = sum(a^x, x=1..inf)
Infinite placeholder did not really get cancelled, but it is OK

sum(a^x * u_x, x, m, n-1) = -a^(x-1) * t * {1 + (tΔ) + (tΔ)^2 + ...} u_x | x = m .. n

sum(2^x * x^2, x, 10, 26-1)
= preval(-2^(x-1) * t * (x^2 + (2x+1)*t + 2*t^2), 10, 26)
= 38788923392 - 67584
= 38788855808

---

More compact formula, for u in falling factorial form:

\(\displaystyle
\sum_{x=0}^{\infty} a^{x+1} \, x^\underline{n} = n! \, t^{n+1}
\quad \text{, where } t = \frac{a}{1-a}\)

\(\sum _{10}^{\infty} 2^x x^2
= 2^9 \sum _{0}^{\infty} 2^{x+1} (x+10)^2
= 2^9 \sum _{0}^{\infty} 2^{x+1} (x^\underline{2} + 21x + 10^2)
→ 2^9 \,t\, (2t^2 + 21t + 10^2)
= -67584\)

\(\sum _{26}^{\infty} 2^x x^2
= 2^{25} \sum _{0}^{\infty} 2^{x+1} (x+26)^2
= 2^{25} \sum _{0}^{\infty} 2^{x+1} (x^\underline{2} + 53x + 26^2)
→ 2^{25} \,t\, (2 t^2 + 53t + 26^2)
= -38788923392\)

Update equivalent formula, with r = 1/t = (1-a)/a = 1/a - 1

\(\displaystyle
\sum_{x=0}^{\infty} \frac{x^\underline{n}}{(r+1)^{x+1}}
= \frac{n!}{r^{n+1} }
\)

[Johnh]

(10-27-2024 04:50 PM)floppy Wrote:  

(10-01-2018 11:34 PM)Albert Chan Wrote:  I "know" the sum does not add up to -1/12, but the guys were so serious ...

The problem is people end by thinking there is a flat earth..

Actually...

A flat earth model is a very good simple model of reality for 90% of what we need a map for! It's ideal for finding your way across the city, or even across the state. But it gradually breaks down as you move to a more 'Global' scale, until if you cling to it for too long, you can prove that by driving off the left side of your map, you will suddenly appear thousands of miles away on the right side of it, and yet you are still in the same place!

It seems to me that number theory, advanced maths, everyday engineering, classical physics and then the most advanced relativity, particle and astro-physics are also on a range of scales where they give good helpful insights within bounds, but if you pursue each one too far, they starts to break down and create absurdities and conundrums. And then we need to move to better, or as yet unknown theories of reality. Or just stick to our day-jobs....

[carey]

(10-27-2024 12:36 PM)AnnoyedOne Wrote:  Which leads me to a few "pet peeves".

"The mass of a photon is zero."
Er, no it isn't. It may be really really small, even unmeasurable, but it has some mass.

As the above is off-topic, my reply will be brief. While everyone is entitled to their personal opinions, there is no debate on this issue in the professional physics community. The Standard Model in particle physics requires a massless photon (gauge symmetry). If the point being made is that it’s not possible to measure zero mass, that’s why increasingly precise physics experiments obtain tighter and tighter upper bounds on the mass, as expected for a massless particle. (To be clear, we’re talking about zero rest mass, not relativistic mass).

Edit: Just noticed that Nigel (UK) made a similar reply.