Maximum fast converging 3 by 2 of pi

3 iterations.3 by 2 of pi to 3 places.

Otherwise we do have 4 exactly.

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Something like this

 An expression like this can be tried for pi .

I hope to get 4 minus pi using golden ratio(3 places)

Another expression to get 4 minus phi using golden ratio (4 places) is 4 -pi = 530/999 times phi

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Got 4

 

I have two solutions currently to get pi to more places.

1. Divide by 4/pi.

2. Increase n in 0.000041206 by 0.0555.

I have watched the Youtube video on pi=4 and this step will require a team, I think.

My fast converging pi formula

9 places of 1 by pi in 1 iteration. 

I am working further on expression like these

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Adds value to my theory

My theory is about swaying hands and legs; three consecutive numbers; four consecutive numbers; and golden ratio. 

This formula augments my theory. 

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Rare and Exact

It is rare to get golden ratio in equations supporting my theory. 

Mein Gott !! Gauss was exactly accurately precise. Can't believe it.  

There are certain things I want to work my thoughts on. Origami sheets, three expressions for pi by 3 root 3. These I am not able to. I am damn too slow in manipulations, calculus. But my understanding is good.

I have ready-made matrix app model. Most of Indian applications and many US applications can utilize this idea. Pre defined matrix model.

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This one too

Blasphemy?

It is a binomial expression which Sir Isaac Newton was an expert in

Missed by a whisker

Gauss constant G can be more easily understood by this one.

A person scores highest possible marks in being fair. And highest possible marks in being unfair. And he is able to do this for ever till infinity. 

Well that's how I see G in this particular expression.  As I see it using Measuring Behavior. 

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Gauss Constant G

I made an original formula for cosh(x).

I can claim G, Gauss's constant as also one in my bag of constants mastered. This is because 

My sigma formula of e to the power x works for e to the power pi into x too.

So cosh(x) consideration for angles is rationally correct. 

G is extremely tough to understand.  It is sheer God genius of Gauss. My three consecutive numbers theory can give G. That's for sure. 

Other constants are 1, root 2, pi, phi, gamma, e. And now G.

Representation of n

Representation of n as 2n!/(2n-1)!-n!/(n-1)! is not a sin. It is practical. 

We do realize that we have 2 legs. We do say 50 steps away. We do hold the cheque with left hand and sign the cheque with right hand. 

Why not 2n -n at the most? Maximum permissible!!

Gamma function is the reason.

This gives half for odd and even

 Here are the expressions

The general expression being

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Hour Glass

 Zero Hour Glass. Equated to zero. This is proper.

All is well that ends well.

Hour glass is not proper.

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An expression for 2

 The following is an expression for 2

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This expression gives half

The following expression gives 1/2 or half.

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My best explanation for 2

 

Wolfram alpha answer when infinity is called 

I feel we must go ahead because

Further explanation

I have been using n or m or x in my expressions. 

What is this n or m or x?

It is this.

Point 1 

Why factorial?

Because I did this experiment. And gamma function. 

Point 2

How 2 in the expression?

This is tough for me. I don't have a convincing answer. Only thing is factorial implies 2 also.

Jai Gauss!!

Gauss as a kid was into triangular numbers a lot. He made many theories . Probably he made this one too.

Three consecutive numbers of my type and its reciprocal both give triangular numbers. 

See below 

It is a decent knock on the door of Abel Prize committee. 

I am thankful to Gauss for that.

The above equation gives weight to my theory as well.

Today Formula

 

Hope you like the above. Have a nice day !!

Pi formulas

Wikipedia formula takes 165 steps for 50 places of pi.

My formula takes 172 steps for 50 places of pi

My this formula takes just 83 steps for 50 places of pi

This is interesting

This equation is interesting 

Have a very nice day, folks !!

Golden Ratio infinite series

Got the infinite series for golden ratio phi. 

Triangular Numbers

An expression which gives triangular numbers is a rational representation because the difference between them is the number system only.

My operation on expression results in difference between two triangular series is rationally correct as they result in number system.

I am talking of expression of the type

Or

So an argument that my expressions do not follow the number system is misleading. 

Two golden ratio equations

These equations do stress the importance of my theory that has been blogged till now.

I wish to state

A raised finger

An unraised finger

The golden ratio relationship 

And consider this

Colin Maclaurin extended the basic 1,x,x^2 triangle and made cosine and sine and other trigonometric functions.  I am inspired by this and am extending golden ratio relationship further in above. What is the harm?

Ramanujan pi is about fast convergence. So are Salamin Brent, Chudnovsky.  They are rich men. They use computers for their convergence. I am a poor man. And trying to get results fast. Maybe rich men like this and give some money.

Pi comparison for 50 places

My formula 172 steps. 

Chudnovsky 3 steps.  Ramanujan 1 6 steps. Ramanujan 2 8 steps. Salamin-Brent 5 steps. 

My type of pi expressions are considered efficient expressions by Wikipedia

 

I earlier tried Chudnovosky type of pi

Comparison of pi formulas

My formula takes 35 iterations to reach 10 places of pi.

Gauss (Salamin and Brent) takes 3 iterations to reach 10 places of pi.

Ramanujan methods (two of them) takes just 1 iteration to reach 10 places of pi.

You can see for yourself. Ramanujan expression converges rapidly. 

I was unable to test Chudonovsky expression. 

About Lagrange

 

I do feel that arithmetic Geometric Mean which was the brainchild of Lagrange "replaces learning about i"

Perhaps three consecutive numbers and four consecutive numbers too.