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Could anyone try to prove that the below conjectured formula is valid for relating $\pi$ with ALL of its convergents - those, which are described in OEIS via A002485(n)/A002486(n)?

$$(-1)^n\cdot(\pi - \text{A002485}(n)/\text{A002486}(n))$$ $$=(|i|\cdot2^j)^{-1} \int_0^1 \big(x^l(1-x)^{2(j+2)}(k+(i+k)x^2)\big)/(1+x^2)\; dx$$ (1)

and in unformatted form:

(-1)^n*(Pi−A002485(n)/A002486(n))=(abs(i)2^j)^(-1)Int((x^l(1-x)^(2(j+2))*(k+(i+k)*x^2))/(1+x^2),x=0...1)

where integer $n = 0,1,2,3,...$ serves as the index for terms in OEIS A002485(n) and A002486(n), and $\{i, j, k, l\}$ are some integers (to be found experimentally or otherwise), which are probably some functions of $n$.

The "interesting" (I think) part of my generalization conjecture is that both "$i$" and "$j$" are present in both denominator of the coefficient in front of the integral and in the body of the integral itself

At this time it could be shown that the formula under question is applicable for some first few convergents (of the A002485(n)/A002486(n) type)

  1. For example for $\frac{22}{7}$

$$\frac{22}{7} - \pi = \int_{0}^{1}\frac{x^4(1-x)^4}{1+x^2}\,\mathrm{d}x$$

with $n=3, i=-1, j=0, k=1, l=4$ - with regards to my above suggested generalization.

In Maple notation

i:=-1; j:=0; k:=1; l:=4;Int(x^l*(1-x)^(2*(j+2))*(k+(k+i)x^2)/((1+x^2)(abs(i)*2^j)),x= 0...1)

yields 22/7 - Pi

  1. It also works for found by Lucas

http://www.math.jmu.edu/~lucassk/Papers/more%20on%20pi.pdf

formula for $\frac{333}{106}$

$$\pi - \frac{333}{106} = \frac{1}{530}\int_{0}^{1}\frac{x^5(1-x)^6(197+462x^2)}{1+x^2}\,\mathrm{d}x$$

with $n=4, i=265, j=1, k=197, l=5$ -with regards to my above suggested generalization.

In Maple notation i:=265; j:=1; k:=197; l:=5;Int(x^l*(1-x)^(2*(j+2))*(k+(k+i)x^2)/((1+x^2)(abs(i)*2^j)),x= 0...1)

yields Pi - 333/106

  1. And it works for Lucas's formula for $\frac{355}{113}$

$$\frac{355}{113} - \pi = \frac{1}{3164}\int_{0}^{1}\frac{(x^8(1-x)^8(25+816x^2)}{(1+x^2)}$$

with $n=5, i=791, j=2, k=25, l=8$ -with regards to my above suggested generalization.

In Maple notation

i:=791; j:=2; k:=25; l:=8;Int(x^(2*(j+2))(1-x)^l(k+(k+i)x^2)/((1+x^2)(abs(i)*2^j)),x= 0...1)

yields 355/113 - Pi

  1. And it works as well for Lucas's formula for $\frac{103993}{33102}$

$$\pi - \frac{103993}{33102} = \frac{1}{755216}\int_{0}^{1}\frac{x^{14}(1-x)^{12}(124360+77159x^2)}{1+x^2}\,\mathrm{d}x$$

with $n=6, i= -47201, j=4, k=124360, l=14$ -with regards to my above suggested generalization.

In Maple notation

i:=-47201; j:=4; k:=124360; l:=14;Int(x^l*(1-x)^(2*(j+2))*(k+(k+i)x^2)/((1+x^2)(abs(i)*2^j)),x= 0...1)

yields Pi - 103993/33102

  1. And also it works Lucas's formula for $\frac{104348}{33215}$

$$\frac{104348}{33215} - \pi = \frac{1}{38544}\int_{0}^{1}\frac{x^{12}(1-x)^{12}(1349-1060x^2)}{1+x^2}\,\mathrm{d}x$$

with $n=7, i= -2409, j=4, k=1349, l=12$ - with regards to my above suggested generalization.

In Maple notation

i:=-2409; j:=4; k:=1349; l:=12;Int(x^l*(1-x)^(2*(j+2))*(k+(k+i)x^2)/((1+x^2)(abs(i)*2^j)),x= 0...1)

yields 104348/33215 - Pi

  1. And it works as well for $\frac{618669248999119}{196928538206400}$

which, by the way, is not part of A002485/A002486 OEIS sequences:

$$\frac{618669248999119}{196928538206400} - \pi = \frac{1}{755216}\int_{0}^{1}\frac{x^{14}(1-x)^{12}(77159+124360x^2)}{1+x^2}\,\mathrm{d}x$$

with $i= 47201, j=4, k=77159, l=14$ -with regards to my above suggested generalization.

In Maple notation

i:=47201; j:=4; k:=77159; l:=14;Int(x^l*(1-x)^(2*(j+2))*(k+(k+i)x^2)/((1+x^2)(abs(i)*2^j)),x= 0...1)

yields 618669248999119/196928538206400 - Pi

This question relates to my answer given in https://math.stackexchange.com/questions/1956/is-there-an-integral-that-proves-pi-333-106/127618#127618

Update: Recently Thomas Baruchel (see his answer at https://math.stackexchange.com/questions/860499/seeking-proof-for-the-formula-relating-pi-with-its-convergents ) has conducted extensive calculations and found that even the parametric formula (with four parameters) yields infinite number of solutions for each "n".

Thomas shared with me his calculations results and supplied me with quite a few of valid combinations of i, j, k, l values - so now I have a lot of experimentally found five-tuples {n,i, j, k, l}, which satisfy above parameterization, where n varies in the range from 2 to 26.

Based on this data, of course, it would be nice to find how (if at all) i, j, k, l are inter-related between each other and with "n" - but such inter-relation (if exists) is not obvious and difficult to derive just by observation ... (though it is clearly seen that an absolute value of "i" is strongly increasing as "n" is growing from 2 to 26). Looking at all available {i,j,k,l} solutions (initial ones found by me and those which were found by Thomas Baruchel program) one could observe that in all of them abs(l-j)=2*m where "m" is some positive integer.

Update #2:

Thanks to Jaume Oliver Lafont, at least one case, answering affirmatively to the last question, is identified: i=-1, j=-2, k=1, l=0

$$\pi = \int_{0}^{1}\frac{4}{1+x^2}\,\mathrm{d}x$$

Should there be infinite number of such cases?

David Trimas looked into the option involving partial solution for my conjecture by setting j=l=0 and deriving formulas for "i" and "k" for j=l=0 condition.

The result is following:

(-1) ^ n * (Pi − A002485(n)/A002486(n)) = ((Abs(i)) * 2 ^ j) ^ (-1) * Int((x ^ l * (1 - x) ^ (2 * (j + 2)) * (k + (i + k) * x ^ 2 ))/(1 + x ^ 2 ), x=0 ...1)

holds true for any n>2 when

i =(-1)^(n) * 3 * A002486(n)

k = (-1)^(n) * (47 * A002486(n) - 15 * A002485(n)

For example for n=3 where A002485(3)=22 and A002486(3)=7

i=(-1)^337=-21

k=(-1)^3*(477 - 1522)=1

and the quick check via Inverse Symbolic Calculator using Maple (also could be done using Mathematica) provides the confirmation.

i:=-21;j:=0;k:=1;l:=0;(abs(i)2^j)^(-1)int((x^l(1-x)^(2(j+2))*(k+(i+k)*x^2))/(1+x^2),x=0...1) = 22/7 - Pi

Substitution of

i =(-1)^(n) * 3 * A002486(n)

and

k = (-1)^(n) * (47 * A002486(n) - 15 * A002485(n)

into general conjectured formula with the condition that j=l=0 (under which above formulas for "i" and "k" were derived) confirmed those formulas for "i" and "k" validity by bringing lhs and rhs to be equal.

P.S. Per discussion with Jaume Oliver Lafont, depending on the value of the polynomial x degree in the integral body's numerator (while denominator stays to be the same "1+x^2"), the result varies from "Pi" to "log(2)" and also to "+/- (Pi - p/q)" as well as to "+/-(log(2)-p/q)", so perhaps now one could produce two distinct families of parameterization: one for Pi and the differences between Pi and its convergents and another for log(2) and the differences between log(2) and its convergents.

As it was mentioned above: "Looking at all available {i,j,k,l} solutions (initial ones found by me and those which were found by Thomas Baruchel program) one could observe that in all of them abs(l-j)=2*m where "m" is some positive integer."

Going further I make another conjecture re identity which relates Log(2) and its convergents and is analogous to the one which relates Pi and its convergents. Below is conjectured by me formula (expressed in Maple notations) for relating Log(2) (that is Ln(2)) with ALL of its convergents - those, which are described via A079942(n)/A079943(n) ratio where A079942(n) and A079943(n) are OEIS integer sequences.

(-1)^n*(Log(2) − A079942(n)/A079943(n))=(Abs(i)2^j)^(-1)Int((x^l(1-x)^(2(j+2))*(k+(i+k)*x^2))/(1+x^2),x=0...1)

where integer n>2 serves as the index for the terms in the OEIS A079942(n) and A079943(n) integer sequences, and {i,j,k,l} are some signed integer parameters (which are some implicit functions of “n” and to be found experimentally or otherwise for each value of “n”) , and abs(l - j) = 2*m + 1, where “m" is some positive integer.

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    $\begingroup$ Possibly somewhat related: mathoverflow.net/questions/67384/… $\endgroup$
    – j.c.
    Jul 10, 2014 at 13:58
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    $\begingroup$ The integral $I_{l,m}=\int_0^1 \frac{x^l(1-x)^m}{1+x^2}\,dx$ is always a linear combination of $1$, $\pi$, and $\log 2$ with rational coefficients. So you can always find $k$ and $i+k$ so that $k I_{l,m}+(i+k)I_{l+2,m}$ is $\alpha+\beta\pi$. I'm not sure how you prove that $\beta/i$ is a power of two, though. $\endgroup$
    – Kirill
    Jul 10, 2014 at 14:34
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    $\begingroup$ It does appear, though, that $x^l(1-x)^m\bmod 1+x^2$ has always zero or a power of two as its constant coefficient. $\endgroup$
    – Kirill
    Jul 10, 2014 at 14:40
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    $\begingroup$ @Kirill, $x^{\ell}(1-x)^m=(1+x^2)p(x)+ax+b$; now evaluate at $x=i$, and note that $1-i=\sqrt2e^{-i\pi/4}$. $\endgroup$ Jul 13, 2014 at 1:07
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    $\begingroup$ May I suggest, Alex, that you learn something about formatting mathematics on this site, as your latest edit is unreadable. Have a look at how the earlier parts of your question are formatted, and/or click on the "help" link on this page. $\endgroup$ Jul 24, 2014 at 0:41

2 Answers 2

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We have

$$(|i|\cdot2^j)^{-1} \int_0^1 \big(x^l(1-x)^{2(j+2)}(k+(i+k)x^2)\big)/(1+x^2)\; dx $$

$$= (\operatorname{sgn}(i) \cdot2^j)^{-1} \int_0^1 \big(x^{l+2} (1-x)^{2(j+2)}\big)/(1+x^2)\; dx +\frac{k}{ |i|\cdot2^j }\int_0^1x^l(1-x)^{2(j+2)}\; dx $$

Now $\frac{1}{2^j }\int_0^1x^l(1-x)^{2(j+2)}\; dx $ is just some rational number so by selecting $k$ and $i$ we can take the second term to be any rational number (positive if $k$ is required to be positive).

So if we evaluate the left term and find it to be $\pm \pi$ plus a rational number, we can deduce that for any other number of the form $\pm \pi$ plus a rational number (with the same sign in the $\pm$) we can choose $k,i$ to make the integral equal that formula.

This is easy to do. If we set

$$\frac{x^{l+2} (1-x)^{2j+2}}{1+x^2} = \frac{a+bx}{x^2} + f(x)$$

for $a,b$ two rational numbers and $f(x)$ a polynomial in $x$ with rational coefficients, then

$$x^{l+2} (1-x)^{2j+4} = a+bx +(1+x)^2 f(x)$$

$$ i ^{l+2} (1-i)^{2j+4} = a + bi$$

$$a+bi = i^{l+2} (-2i)^{j+2} = 2^2 \cdot 2^j \cdot i^{l-j} $$

As long as $j$ or $l$ are congruent modulo $2$, this is $\pm 4 \cdot 2^j$, so the leftmost term is the integral of a polynomial with rational coefficients plus

$$ \pm 4 \int_0^1 \frac{1}{1+x^2} dx = \pm \pi$$ as desired.

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  • $\begingroup$ Underrated answer imo. Easy and short but well explained and efficient. +1 $\endgroup$
    – mick
    Jun 21, 2023 at 21:18
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Here's the proof of my conjecture developed by Ute Hahn: https://www.quora.com/profile/Alexander-R-Povolotsky/p-121330821?ch=15&oid=121330821&share=8c83048d&srid=u1RAda&target_type=post

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