Geometric Model of Walker's Equation and Walker's Series !

[Commander]

Hi I give here the Geometric Model and representation of Walker's Equation and proposal of an Equation od Infinite Series which I call 'Walker's Series' which appears to have escaped many !

Proof of Walker's equation.docx - Google D

Dear Moderator,

I tried to remove the duplicate images in this post but couldn't succeed

Plz remove the repeating pics TY !

Edited August 20 by Commander

[Commander]

1 = 1/(n+1) +  n/(n+1)(n+2) + n/(n+2)(n+3) + n/(n+3)(n+4) ….  etc till infinity 

For any n (positive integer) from 1 to infinity 

This may be called  Walker’s Series’
By Wg Cdr Thomas Walker - 
22 Aug 2024 

FYI plz

Edited August 22 by Commander

[joigus]

Clever.

But,

the way in which you sub-divide the side of the square is divergent, as,

\[ \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \cdots \]

is the well-known harmonic series  which is divergent. So,

\[ \left( 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \cdots \right)^{2} \]

cannot possibly give you a convergent series, I would say. This is compounded with the fact that what you have on your RHS is an infinite series of infinite series.

Sometimes it happens that a divergent series can be useful because it can be regularised, or made sense of in some clever way. Euler was a master at this. Have you tried to discuss it with a professional mathematician?

By the way, that would be an identity, not an equation. Otherwise, what is the unknown to solve for?

Edited August 22 by joigus
Latex editing

[Commander]

Hi

Yes Harmonic Series is Divergent but Walker’s Series is not divergent.

No member of the Series has a value of more than 1/(n+1) and no FACTOR used in the Equation will be less than ‘n’ in value either in Denominator or Numerator.

 

1 = n/n(n+1) + n/(n+1)()n+2) + n/(n+2)(n+3) + …….. 

 

If n = 1 :  then 1/1x2 + 1/2x3 + 1/3x4 +  ………

As n increases

1= 2/2x3 + 2/3x4 + 2/4x5  + ……….

Similarly

1 = 3/3x4 + 3/4x5 + 3/5x6  + …….

1 = 4/4x5 + 4/5x6 + 4/6x7  + ………..

.

.

 

.

[joigus]

Ok. You haven't provided any proof of convergence yet.

On the LHS you have the square of a divergent series. So that bit certainly cannot be equated to 1.

On the RHS you have an infinite sum of different convergent series. Taken one by one, they are all convergent (as per comparison test), as far as I can see. But, mind you, you have an infinite sum of infinite sums! 

I think you may have found an interesting relation, which I would call "improper identity"? Certainly, not an equation. Sometimes, divergent series, upon further examination, can be found to be quite interesting, perhaps through a singularity or pole of a well-known function, etc. One famous example is the improper identification 1+2+3+... = -1/12. These identites rarely mean what they say; they mean something rather more abstract and sophisticated.

Professional mathematicians are experts at getting robust proofs from arguments like this. Why don't you try getting in touch with some expert in analysis in academia?

As to originallity, don't put too much stock in it. It is said that every discovery has been discovered before. And please, do not name it after yourself. That's frowned upon in the academic world.

[Commander]

Hi Joigus

TY

Earlier I had given the proof of Walker’s Equation and now I have depicted the Proof Geometrically as can been seen both for Summing upto 1 and converging 

The same depiction also indictes the Geometric representation of Walker’s Series for those who can visualise !

The General form of Walker’s Series for n from 1 to infinity can also be inferred from this !

If I write up more on this I will put up here !

 

 

We can see 1/n = 1/(n+1) + 1/(n+1)^2 + 1/(n+1)^3 + …… as per Walker’s Equation Proof Steps.

Which is : 1/n - 1/n+1 = inverse powers of (n+1) from 2 to infinity as shown above which can be called RHS

The LHS is 1/n - 1/n+1 = 1/n(n+1)

And this progression goes on ……..

The LHS a Linear sum of Factors adds up to 1 and the RHS a double matrix depiction of inverse powers adds up to 1 too !

The LHS gives rise to the Walker’s Series and the RHS gives rge Walker’s Equation !

 

 

Hi

Yes Harmonic Series is Divergent but Walker’s Series is not divergent.

No member of the Series has a value of more than 1/(n+1) and no FACTOR used in the Equation will be less than ‘n’ in value either in Denominator or Numerator.

 

1 = n/n(n+1) + n/(n+1)()n+2) + n/(n+2)(n+3) + …….. 

 

If n = 1 :  then 1/1x2 + 1/2x3 + 1/3x4 +  ………

As n increases

1= 2/2x3 + 2/3x4 + 2/4x5  + ……….

Similarly

1 = 3/3x4 + 3/4x5 + 3/5x6  + …….

1 = 4/4x5 + 4/5x6 + 4/6x7  + ………..

.

.

 

.

53 minutes ago, Commander said:

Hi Joigus

TY

Earlier I had given the proof of Walker’s Equation and now I have depicted the Proof Geometrically as can been seen both for Summing upto 1 and converging 

The same depiction also indictes the Geometric representation of Walker’s Series for those who can visualise !

The General form of Walker’s Series for n from 1 to infinity can also be inferred from this !

If I write up more on this I will put up here !

 

We can see 1/n = 1/(n+1) + 1/(n+1)^2 + 1/(n+1)^3 + …… as per Walker’s Equation Proof Steps.

Which is : 1/n - 1/n+1 = Sum of inverse powers of (n+1) from 2 to infinity as shown above which can be called RHS

The LHS is 1/n - 1/n+1 = 1/n(n+1)

And this progression goes on ……..

The LHS a Linear sum of Factors adds up to 1 and the RHS a double matrix depiction of inverse powers adds up to 1 too !

The LHS gives rise to the Walker’s Series and the RHS gives rise Walker’s Equation !

 

 

 

 

Edited Tuesday at 05:58 AM by Commander

[Commander]

1 hour ago, Commander said:

Hi Joigus

TY

Earlier I had given the proof of Walker’s Equation and now I have depicted the Proof Geometrically as can been seen both for Summing upto 1 and converging 

The same depiction also indictes the Geometric representation of Walker’s Series for those who can visualise !

The General form of Walker’s Series for n from 1 to infinity can also be inferred from this !

If I write up more on this I will put up here !

 

We can see 1/n = 1/(n+1) + 1/(n+1)^2 + 1/(n+1)^3 + …… as per Walker’s Equation Proof Steps.

Which is : 1/n - 1/n+1 = Sum of inverse powers of (n+1) from 2 to infinity as shown above which can be called RHS

The LHS is 1/n - 1/n+1 = 1/n(n+1)

And this progression goes on ……..

The LHS a Linear sum of Factors adds up to 1 and the RHS a double matrix depiction of inverse powers adds up to 1 too !

The LHS gives rise to the Walker’s Series and the RHS gives rise to Walker’s Equation !

 

.

.

 

.

 

 

Edited Tuesday at 06:20 AM by Commander

[joigus]

5 hours ago, Commander said:

Earlier I had given the proof of Walker’s Equation [...]

When?

5 hours ago, Commander said:

If I write up more on this I will put up here !

How about proving convergence?

5 hours ago, Commander said:

We can see 1/n = 1/(n+1) + 1/(n+1)^2 + 1/(n+1)^3 + …… as per Walker’s Equation Proof Steps.

No. We can't see that because that doesn't make any sense. 1/n is a term of the harmonic series, while 1/(n+1)+1/(n+1)²+1/(n+1)³+... is an infinite series. Therefore, the RHS either diverges or is a number, and has no n-dependence. You're saying that \( 1/n = \pi²/6 -1  \) (see below: Basel problem).

5 hours ago, Commander said:

Which is : 1/n - 1/n+1 = inverse powers of (n+1) from 2 to infinity as shown above which can be called RHS

No. What you're doing here is use the partial decomposition trick,

\[ \frac{1}{n\left(n+1\right)}=\frac{1}{n}-\frac{1}{n+1} \]

So you've split a convergent series as the difference of two divergent series. On the LHS you have the famous series in the Basel problem, which converges to \( \pi²/6 \) so what you're saying is,

\[ \frac{\pi²}{6}=\infty-\infty \]

which, of course, is totally meaningless.

 

I meant PFD (partial fraction decomposition) before.

[joigus]

2 hours ago, joigus said:

No. We can't see that because that doesn't make any sense. 1/n is a term of the harmonic series, while 1/(n+1)+1/(n+1)²+1/(n+1)³+... is an infinite series. Therefore, the RHS either diverges or is a number, and has no n-dependence. You're saying that 1/n=π²/6−1 (see below: Basel problem).

Sorry, I made a mistake here. Both sides have n-dependence. I'll get back asap. 

[joigus]

Ok, I went back to your square, but on second thought I can't make sense of it. I thought I understood (crudely) what you were trying to do. Now I see you're dividing the square into pieces that actually overlap, so it's not a partition of the square really. It's something else.

So I went to a completely analytical POV, ignoring the picture.

I see no proof of convergence yet. Maybe you provided it before on some other thread, but I missed it.

In purely analytical language, what you're saying is that,

\[ \sum_{n,m=2}^{\infty}\frac{1}{n^{m}}=1 \]

which, yes you're right can be proven, as the partial sum satisfies,

\[ \sum_{n=2}^{k}\sum_{m=2}^{\infty}\frac{1}{n^{m}}=1-\frac{1}{k} \]

So, yes, you're absolutely right AFAICT. But I still don't understand your square, I'm sorry. I had to interpret it purely analitically, with no pictures.

 

 

 

 

 

PS: I tried to relate it to Riemann's zeta function, but I fell back to an infinity-infinity indeterminate,

\[ \sum_{n,m=2}^{\infty}\frac{1}{n^{m}}=\sum_{m=2}^{\infty}\left(\zeta\left(m\right)-1\right) \]

as I told you your method seems to indicate. The way to go is to build the partial sum

Anyway... I'm a bit tired to do hard math now. That was fun.

[Commander]

Walker's Equation by Wg CdrThomas Walker.pdfWgCdrThomasWalkersTheories.pdfWalker's Equation.docWalkerEqn Jim Loy.docACuriousConnection.pdf

Now with these basic Structures indicated I believe that Logically , Algebraically and by Mathematical Induction we can expand these equations to explore and detect other Relations involving numbers other than Integers !

BTW

Some of these old documents may contain my old Contact details

My present address 

Flat # 003

Sai Excellency Apartments 

2nd Cross Hennur main road 

Bangalore 560043

India

Phone 9880184818 and 8884577768 both having WhatsApp

Edited Wednesday at 04:02 AM by Commander
Spelling

[Commander]

This is some recognition I got in a paper when I was putting up my findings in some University websites

[Commander]

Let’s take a look into the Geometrical Model which has horizontal lines drawn at ½, ⅓, ¼ etc upto infinity

The area below each line is same value inn sq units that is ½,⅓,¼, etc

Each one of these are also equal to 1/1x2 , 2/2x3, 3/3x4, etc

And each has a power expansion of (⅓ + ⅓^2 +⅓^3 …) , (¼ + ¼^2 + ¼^3 ….) . (⅕ + ⅕^2 +⅕^3 ….. )

 

And now.

The area below each line is the sum of all rectangles below it : 

That is

½ = 1/2x3 + 1/3x4 + 1/4x5 ……

⅓ = 1/3x4 + 1/4x5 + 1/5x6 …….

¼ = 1/4x5 +1/5x6 + 1/6x7 ……..

Etc

 

And therefore the Walker’s Series is proved as

1 = 2/2x3 + 2/3x4 + 2/4x5 ……..

1 = 3/3x4 +3/4x5 + 3/5x6 ………

1 = 4/4x5 + 4/5x6 + 4/6x7 ………

 

By Wg Cdr Thomas Walker 

Bangalore India

11 Sep 24

[joigus]

Ok. Thank you. That's a tad more information than I needed. Unfortunately I won't be able to pay you a visit any time soon.  

I'll take a look at the most significant bits when I get the time.

Also, if you don't mind my saying, I would advise you to lower down a bit your expectations of getting credit. Most of these series have been summed and understood centuries ago. 

[Commander]

We see that while

 

1 = ½ + 1/2x3 + 1/3x4 + 1/4x5 ………

It also validly sums up for every n terms 

Ie

 

1 = ½ + 1/2      for n=2    

1 = ½ + 1/2x3 + ⅓       for  n=3 

1 = ½ + 1/2x3 + 1/3x4 + ¼  for n=4

1 = ½ + 1/2x3 + 1/3x4 + 1/4x5 + ⅕  for n=5

etc

OK TY

I understood what you said and what you mean !

Thanks !

I can make out many explanations are a bit difficult to understand from the depiction and notes but I can vouch I can explain it more clearly if someone wants to know !

Edited Thursday at 05:30 AM by Commander

[Commander]

To Summarize : 

 

1 = 1/1x2 + 1/2x3 + 1/3x4 + ……. + 1/(n-1) x n  + 1/n(n+1) + 1/(n+1)(n+2) + 1/(n+2)(n+3) + ……….

1 - (1/1x2 + 1/2x3 + 1/3x4 + ……. + 1/(n-1) x n ) =  1/n(n+1) + 1/(n+1)(n+2) + 1/(n+2)(n+3) + ……….

Which is

1/n =  1/n(n+1) + 1/(n+1)(n+2) + 1/(n+2)(n+3) + ……….

Therefore

1 =  n/n(n+1) + n/(n+1)(n+2) + n/(n+2)(n+3) + ……….

OR

1 = 1/(n+1) + n/(n+1)(n+2) + n/(n+2)(n+3) + ……….

 

13 Sep 2024