Higher approximations of Ramanujan formula for ##\pi##

[exponent137]

TL;DR Summary

Is it known about the Ramanujan formula for $\pi$ either it has higher approximations or it does not have?

One of the Ramanujan formulae for π is
$(9^2+22^2/19)^{1/4}=3.14159265258$.
It is precise to 9 digits. I did not read about higher approximations so that they are by the same pattern and that describe an arbitrary number of digits of π. Do these higher approximations not exist, or it is not known whether they exist?

One publication of this formula is in http://ramanujan.sirinudi.org/Volumes/published/ram06.pdf on page 43.

Other links about Ramanujan can be found on Google, for instance:
https://en.wikipedia.org/wiki/Srinivasa_Ramanujan

 

[fresh_42]

In 1989, the Chudnovsky brothers computed π to over 1 billion decimal places on the supercomputer IBM 3090 using the following variation of Ramanujan's infinite series of π:

https://en.wikipedia.org/wiki/Approximations_of_π#20th_and_21st_centuries

 

[exponent137]

In 1989, the Chudnovsky brothers computed π to over 1 billion decimal places on the supercomputer IBM 3090 using the following variation of Ramanujan's infinite series of π:

View attachment 304430https://en.wikipedia.org/wiki/Approximations_of_π#20th_and_21st_centuries

Yes, this your approximation is infinitely precise, as the distinction of my mentioned formula.

But can my formula above be the beginning of an infinitely precise formula? Or is it known that it cannot be?

 

[fresh_42]

An infinitely precise formula is the definition by series, e.g. $\pi = 4 \displaystyle{\sum_{n=0}^\infty \dfrac{(-1)^n}{2n+1}}.$

If you ask, if there is a series $(f_n)$ of formulas where $f_n$ is precise to $n$ digits, then yes:
$$
f_n:=[\pi \cdot 10^{n-1}]
$$

 

[exponent137]

Yes, but your two series do not include my formula in any step. My question is: does exist a series that includes my mentioned formula?

 

[fresh_42]

Yes, but your two series do not include my formula in any step. My question is: does exist a series that includes my mentioned formula?

Sure.
$$
3\, , \,\left[\dfrac{31}{10}\right]\, , \,\left[\dfrac{314}{100}\right]\, , \,\left[\dfrac{3141}{1000}\right]\, , \,\left[\dfrac{31415}{10000}\right]\, , \,\left[\dfrac{314159}{100000}\right]\, , \,\left[\dfrac{3141592}{1000000}\right]\, , \,\left[\dfrac{31415926}{10000000}\right]\, , \,(9^2+22^2/19)^{1/4}
\, , \,\left[\dfrac{3141592653}{1000000000}\right]\, , \,\ldots
$$

This should demonstrate the weakness of your question. There is no meaningful answer. I gave you already three different explanations and all of them are correct answers to your question.

 

[exponent137]

This your infinite series is written with infinite number of characters. I wish that such a series is written with a finite number of characters, such as your other two formulae for $\pi$. But it should also include my formula as one of the first steps.

I expect answers such as: 1) it is possible but we do not know. 2) It is not possible. 3) It already exists. But I expect that only answer 1) is real?

Here it is also a problem with language to define this problem and communicate it.

 

[fresh_42]

I expect answers such as: 1) it is possible but we do not know. 2) It is not possible. 3) It already exists. But I expect that only answer 1) is real?

And I ask you to define "it" in a precise way. You adjust "it" with every answer of mine.

 

[exponent137]

Yes, I knew my definition from the beginning. But here is the problem with language. If I write it in a precise way, there is a problem that no one will understand it. And English is not my first language. And also I am not familiar with precise mathematical language and definitions.

 

[fresh_42]

Yes, I knew my definition from the beginning. But here is the problem with language. If I write it in a precise way, there is a problem that no one will understand it. And English is not my first language. And also I am not familiar with precise mathematical language and definitions.

You could try https://translate.google.com/. It usually produces acceptable results. I don't think it is a matter of language. You used a formula and want to generalize it. What is a formula? What makes our computation different from my quotients? Both are formulas. What is an allowed generalization and what is not? It is not easy to make your question rigorous, if not impossible. You said that Chudnovskys' formula uses an infinite series. This is true, but we could stop the addition at any finite point and still get a fair amount of correct digits. Just choose the upper limit of the sum big enough.

 

[fresh_42]

$$
\pi \approx \dfrac{1}{12 \displaystyle{\sum_{k=0}^{10^{(10^n)}} \dfrac{(-1)^k(6k)!(13561409+545140134\,k)}{(3k)!(k!)^3 640320^{3k+3/2}}} }
$$
should be a formula that yields $n$ correct digits, at least for $n$ up to a billion.

 

[exponent137]

I wish to estimate whether my above formula of Ramanujan is accidental, or it is a part of some absolute precise series for $\pi$. Namely, it is very accurate and very simple.

I see that you know the formulae of Ramanujan a lot, do you know any other reference with analysis for my mentioned formula, except my mentioned reference in the first post?

fresh_42: You adjust "it" with every answer of mine.
I wrote in my first post: "so that they are by the same pattern" This meant that I wished for a different type of formula than yours mentioned in the post from Yesterday, 11:26 AM

What I wrote is similar to convergence:
https://en.wikipedia.org/wiki/Convergent_series
the distinction is that here also the length of a record of a formula is convergent. It is not convergent in your post from 11:26 AM.

 

[exponent137]

Summary: Is it known about the Ramanujan formula for $\pi$ either it has higher approximations or it does not have?

One of the Ramanujan formulae for π is
$(9^2+22^2/19)^{1/4}=3.14159265258$.
It is precise to 9 digits. I did not read about higher approximations so that they are by the same pattern and that describe an arbitrary number of digits of π. Do these higher approximations not exist, or it is not known whether they exist?

One publication of this formula is in http://ramanujan.sirinudi.org/Volumes/published/ram06.pdf on page 43.

Other links about Ramanujan can be found on Google, for instance:
https://en.wikipedia.org/wiki/Srinivasa_Ramanujan

I made a lapse in the Ramanujan formula. It should be written as :
$(9^2+19^2/22)^{1/4}=3.14159265258$.

 

[fresh_42]

Here is a formula from Ramanujan that looks far more promising:
https://de.wikipedia.org/wiki/Kreiszahl#Numerische_Verfahren_ab_dem_20._Jahrhundert

It is what you are asking for, a limit. It gets more and more precise the more summands we add. Of course, $\pi$ is not a rational multiple of $\sqrt{2}$ so it won't ever become equal, but arbitrary close.

Your formula is only one irrational number. Even your source says ...

This value was obtained empirically, and it has no connection with the preceding theory.

... that this specific number is not part of any procedure and only a lucky punch.

 

[exponent137]

Here is a formula from Ramanujan that looks far more promising:
https://de.wikipedia.org/wiki/Kreiszahl#Numerische_Verfahren_ab_dem_20._Jahrhundert

You wrote "promising" for formula in section "Numerische Verfahren ab dem 20. Jahrhundert".
Is it not yet 100% sure, that it is limiting toward $\pi$?

In this link, Euler's formula is also evident:
$\frac{\pi^4}{90}=\frac{16}{15}\times...$
If we modify it to:
$\frac{\pi^4}{90}=\frac{13}{12}\times...$,
we obtain my mentioned Ramanujan Formula without 1/11.
(In full it can also be written as: $\pi^4\approx 97.5-1/11$)

Your formula is only one irrational number. Even your source says ...

... that this specific number is not part of any procedure and only a lucky punch.

Yes, this was only a lucky punch for Ramanujan. But it is possible that one day someone or some AI software will find a generalized formula of it that will limit toward $\pi$.

It is also interesting that the formula for $\pi^8/9450$ at the term $11^8$ becomes more precise than my mentioned Ramanujan formula.
It is also interesting that $9450=42\times 15^2$. If we modify it to $(42+1/4)\times 15^2=97.5^2$ we obtain the above mentioned number 97.5. And of course, this modification should be explained, as well as the above modification to $13/12$. Thus, lucky similarities exist, but the ultimate explanation for these similarities does not exist yet.Additional question: In your link, there are two formulae for $\pi^2/6$. What is the relation between them?