A rare 1 by pi set of expressions have been obtained. The exact relationship between k and a is not obvious. I need to check more to make a much more compact formula.
Readers may check this set of expressions and post their comments below.
My sincere prayers for peace !!
There is a pattern to the earlier expression in my last blog post.
And that is
Hope you have a nice day !!
I have made this simple expressions.
You may try them out in Wolfram Alpha or Symbolab or other sites. You may extend this simple expression and tell me in this blog comments if you find a pattern.
My good wishes to you. Have a nice day !!
It started in 2010. When I accidentally made an infinite series. Measuring Behavior.
Then I began to focus.
My posting in R&D was a surprise. But was a blessing to extend my infinite series thoughts. I was alone in Ranchi. Hitler bunker psyche. Hitler a scientist? You bet !! My claim.
Anyway that is another story.
I was asked to write a paper. I realized it has hyperlink trees properties. Greatest scientists and engineers are involved. My mathematics though had only one reference. Wikipedia. Jetir were my main publishers of my mathematics expressions.
An infinite Geometric progression, pentagon and Euler-Mascheroni thoughts were the first in mathematics in Ranchi.
My desire was high. I did not get pi constant. I had Finch ebook on Mathematical constants. I wanted to realize all the major constants. Even Kinchin constant. All nature blessed constants. Perhaps Feigenbaum constant is of nature too. I have never put a thought to it.
So that is what it is.
I possibly should have referred to my Rational Series as Gauss extension of triangular numbers. That might have forced the major mathematical journals to accept my papers. Because they cannot ignore my claim for Gauss. And today I might be in Chennai Mathematical Institute.
So what I have? 1, root 2, golden ratio, e, pi, gamma, Apery constant, Catalan constant. In some cases I have more than one expressions for these constants.
And I also have nothing. ( Tata Steel!!)
I have made a decently beautiful binomial pattern.
Though I made similar binomial expressions earlier, this one is a little different.
Hope you like it !!
Here are the expressions.
I have worked on Rational number series or what I have termed as consecutive number theory a lot for the last three years.
I have come up with a myriad of expressions simply by tweaking around with the individual terms in the expressions.
I have continued it further today.
I have made a binomial expression. Here it is
Hope you have a nice day !!
Here are the screenshots.
Hope you enjoy a relaxed Sunday!!
Below I have re-done the Madhava of Sangamagrama pi by 4 formula and Basel Problem in the three consecutive number model.
Hope you like these ideas of mine.
Have a great weekend !! Indulge and enjoy !!
A little tweak of a special type on three consecutive numbers and it is great as Dirichlet beta function.
And I have used it below to get an expression of the Catalan constant.
So all in all with the consecutive numbers model, I have managed to get expressions for 1, root 2, pi, e, golden ratio, Euler Mascheroni constant, Apery Constant and Catalan constant.
Have a nice day !!
A little tweak in the three consecutive numbers model and it is great as Riemann zeta function.
And I have used it below to get the Apery constant.
Hope you all like it.
Have a very nice day !!
Euler-Mascheroni constant or gamma is also obtained from the consecutive number theory.
Here is a screenshot.
Hope you like it.
Have a very nice day !!
Observe how alternate negation changes hyperbolic to trigonometric functions. I guess the movie "Anantaram" by Adoor Gopalakrishnan is somewhat about this. Not this formula. But Madhava of Sangamagrama 's. You look at one way the main character of the movie is empathized. Look another way the family is. And in the end the flashback of the young kid steps down into pond alternatively. It is quite a good movie. So is "Mithilukal". And "Nizhalkuthhu". I strongly recommend these movies of Adoor Gopalakrishnan to my foreign blog readers.
Coming back to hyperbolic and trigonometric formulas, here are they.
Got it !! At last !!
A golden ratio expression with consecutive number theory. Here it is.
This is indeed a very happy day for me.
Hope you all have a very nice day !!
I am not showing off. But I am confident. It is a situation which Kapil paaji faced in finals at 1983 in Lords. You need to perform. Not just claim your place. Yes, I can perform.
My best wishes to all my blog readers !!
I want to read Gauss. I have a lifetime ahead to do that.
My desire to learn mathematics was always about focus. Earlier about two years back I wanted to learn mathematical analysis, Gauss, Euler, Abel and Ramanujan. Then till about three months back it was Gauss and Euler. And presently it is just Gauss.
Gauss was out of this world. If you consider a three dimensional normal distribution mountain of mathematicians, the only person who is at the top of the mount is Gauss.
It may cost me 5 lakh rupees (a considerable amount) to buy all the necessary books on Gauss. It will be easy if I get a couple of library cards of IISc or IIT. But these institutions are away from Ranchi. I tried the online library of IIT Kharagpur, but it seemed to be of not much help over internet.
The world is moving ahead so fast. Electric cars, settlement in Mars. I don't know how 2023 or 2024 will be. India should not lag behind. Electric buses, electric metros, fast trains, quality food, super fast internet, smart cities all should be our very near objectives. I also read about the steel used in electric cars of Europe, USA and China. India should take up manufacturing this steel. R&D efforts should be initiated. This steel it seems can generate electric and magnetic fields in many axes. And that is why an engine the size of two coca cola cans can develop about 136 HP power. Maruti SX4 has 104 HP power.
Come on India. We can make it. It is not just a fight against Corona. It is about making a mark in future time.
This one is similar to Ramanujan and Chudnovsky formulas.
This gives pi to 12 places in 6 iterations.
Wishes for a very nice day!!
The below expressions are beautiful.
Start from 1 you get 0.
Start from 0 you get 1.
Hope you have a nice day !!
I have made a formula for 1 by pi to 20 places. 6 iterations.
Hope you like it.
Have a nice day !!
Ramanujan was deeply inspired. He was religious. Professor Hardy was moved by his ideas and brought him to Cambridge University. Ramanujan is famous for Ramanujan Sato series, Ramanujan Tau functions, Ramanujan theta functions. Two constants are named after him. He has many many beautiful expressions which stimulates interest to mathematicians. He has lot of theories which are puzzling even today.
Harish-Chandra was a great guy. He had a lot to say in mathematics in terms of expressions. He was damn good when I see his work in Wikipedia. He was unlucky to miss the Fields medal in 1958.
Madhava of Sangamagrama was a brilliant man. He made pi by 4 formula which is called Madhava- Leibniz pi formula. He made a lot of contributions to circular functions.
This is the suggested pi formula for fast convergence and more precision.
One is the seed and the other is iterative. I am not fully sure whether this will work. A big computer is needed for testing the viability.
Let's face it. It is Gauss.
They were of different generations. Euler was earlier to Gauss. Euler died in 1783. Gauss was born in 1777.
I think it is the genius of Gauss which invites opponents.
Ramanujan was a very simple man. Gentleman. Genius. God gifted. There are many theories he made which even today mathematicians are puzzled about. He possibly saw numbers in a special way. And he worked so hard.
Fetches 1 by pi to 16 terms in 4 iterations.
Added
Years back, I did this 1 by pi. This one is fast. 9 places in 4 iterations.
I took (1+0.000041206)^n instead of (1+(0.000041206)^n) as I have been doing.
I am sorry if people are confused.
