What is 1+2+3+4+5+...?

[GaryProtein]

.
Infinity is fine for mathematicians. They can deal with abstractions.

Infinity is a nightmare for physicists. They have to deal with the concrete world.



question for computer people or mathematicians:

Can a presently available computer deal with numbers as great as ten to the billionth power as a possible substitute for infinity?

 

[Mike]

The calculation
2*S2 is meaningless.
S2 = 1 - 2 + 3 - 4 + 5 - 6 +7
> - 2 + 3 - 4 + 5 -…..
= 1 + 1 + 1 + 1 + …., which doesn't converge.
You can't then add (or subtract) two things that don't converge: there is not arithmetic infinity + infinity. Arithmetic only makes sense for Natural numbers. If you want to adjoin a number "infinity" to your Natural numbers, you need to define the arithmetic of these new numbers.
Also, you can't divide by 0, as 0 doesn't have an inverse: there doesn't exist a number 0^-1 such that 0*0^-1 = 1, etc.

 

[GaryProtein]

^^^^ Is that to say the sum of 1+2+3+4+5. . . . is undefined ?

 

[ack]

Found this article about it:

https://plus.maths.org/content/infinity-or-just-112

A lot of people have attacked the proof iin the video by attacking their claim/assumption that 1-1+1-1+1-.... -> 1/2. However, as always, things are not so simple. The original video is a short version of other much more elaborate proofs that your article simply failed to acknowledge. For example, in my thread about all this over a year ago, I posted a video specifically about this 1-1+1-1... sequence. To make a long story short, this sequence is a sequence between two states, +1 and -1. Now imagine these two numbers representing Light On and Light Off states. Then one asks: over time, is the sum of Light On + Light Off+... really equal to Light On or Light Off??? Neither; clearly over time the light is not always on, nor is it off... it's in some intermediate state, and clearly light DOES shine to some degree, over time... That's where the "1/2" comes from, in layman's terms; this series is called Grandi'sSeries, and you can see the proof to convergence to 1/2 at https://en.wikipedia.org/wiki/Grandi's_series . The bottom line is that every attack to the proof in the original video has thusfar been too shallow; this is why the series and the result of -1/12 has been taken so seriously in string theory, because it's not a trick, but a *possible* and convenient result.

BTW, the Reimann zeta function is something we also discussed back then, and it's a basic tenet of technical computing software as well; i.e. real...

 

[Mike]

^^^^ Is that to say the sum of 1+2+3+4+5. . . . is undefined ?

It doesn't converge, or, increases without bound; so, it isn't a Natural number (or a Real number, etc.).

 

[amirm]

The calculation
2*S2 is meaningless.
S2 = 1 - 2 + 3 - 4 + 5 - 6 +7
> - 2 + 3 - 4 + 5 -…..
= 1 + 1 + 1 + 1 + …., which doesn't converge.
You can't then add (or subtract) two things that don't converge: there is not arithmetic infinity + infinity.

And that is precisely what Riemann challenged. He created a zeta function which ack mentions:

For values of S> 1 those become smaller and smaller fractions so one can imagine them having a limit. Reimann's genius was to say, why not allow the function to still operate for values where the sum "blows up" seemingly to infinity?

That was not too much of a reach because the function worked for imaginary numbers and we know those "don't exist." Pictorially, this is what it looks like:

The X axis is real numbers, Y axis is imaginary.

Reimann mathematically proved that there are real numbers representing all values other than when s = 1. If for example we plug -1 in for s, we get our 1+2+3+4+5 sequence which has the answer of... -1/12.

As I mentioned, the first giant leap here is accepting that imaginary numbers work. If we are able to swallow that, then this extension is not all that different.

BTW, there is a million dollar prize for anyone who proves or disproves that for all values where the zeta function results in zero, they all lie along a vertical line at x = 1/2:

A lot of those values have been found but so far, no proof. Reiman's hypothesis (i.e. where all the non-trivial zeros exist) is nevertheless uses to solve other problems.

 

[Mike]

And that is precisely what Riemann challenged. He created a zeta function which ack mentions:

For values of S> 1 those become smaller and smaller fractions so one can imagine them having a limit. Reimann's genius was to say, why not allow the function to still operate for values where the sum "blows up" seemingly to infinity?

That was not too much of a reach because the function worked for imaginary numbers and we know those "don't exist." Pictorially, this is what it looks like:

The X axis is real numbers, Y axis is imaginary.

Reimann mathematically proved that there are real numbers representing all values other than when s = 1. If for example we plug -1 in for s, we get our 1+2+3+4+5 sequence which has the answer of... -1/12.

As I mentioned, the first giant leap here is accepting that imaginary numbers work. If we are able to swallow that, then this extension is not all that different.

BTW, there is a million dollar prize for anyone who proves or disproves that for all values where the zeta function results in zero, they all lie along a vertical line at x = 1/2:

A lot of those values have been found but so far, no proof. Reiman's hypothesis (i.e. where all the non-trivial zeros exist) is nevertheless uses to solve other problems.

Hi Amir,

I'm familiar with Riemann and the RH. I believe that it was Pierre Deligne that got his Field's Medal for his work on this. It's been about 20 years since I read anything that he did

I'm not sure what you mean by "they don't exist". Complex numbers do exist; they're no different from other elements of any other field.

Making a statement about a series of what are obviously rational numbers and making the same statement about complex numbers (which form a Field, unlike the Natural numbers) or elements from other fields (I think that this is what the work of Deligne was about) is a little confusing. 1 -2 + 3 -4 + 5 -6 + 7, etc. doesn't converge in the real numbers. Any way you write it as a sequence of finite terms doesn't form a Cauchy sequence, so it can't converge.

The point that Ack mentioned about states, where -1/2 is the "average," kind of reminded me of throwing a die, where the expected values is 3.5 - not an observed vale.

 

[ack]

I really like MasterChief's link, despite the fact it fails to disprove the proof. And I like it because a) it links to a NYT article which correctly says that -1/12 has been proven in lab experiments to "many decimal places"; one of which is b) the Casimir effect calculations in his article:

Take a moment to take this in. Quantum physics says the energy density should be

That’s nonsense [ack: there isn't infinite energy between the two plates], but experiments show that if you (wrongly) regard this sum as the zeta function

evaluated at

, you get the correct answer. So it seems that nature has followed the ideas we explained above. It extended the Euler zeta function to include values for

that are less than 1, by cleverly subtracting infinity, and so came up with a finite value. That’s remarkable!

Those laboratory verifications is what's really fascinating about the whole thing.

 

[Recordsguy]

Yes since it is the negative of your post count with digits transposed .

I feel so smart that I may go out and invent something[smile]!!

 

[amirm]

I feel so smart that I may go out and invent something[smile]!!

Let's do it together. That way we will assure it will have no useful utility.