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Test this assertion correctly? (A rational approximation to $\pi$?)

Posted 3 months ago
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In A Rational Approach to Pi Beukers asserts on the right column of page 377 that the integral, $$I_n=\int_{0}^{\pi/4}d\phi \Big( 4 \sin(\phi) \big(\sin(\phi)-\cos(\phi)\big) \Big)^n,$$
Produces a rational approximation to $\pi$, which tends to quality $Q$ greater than $0.90$ in the limit of $n \rightarrow \infty$.

To test this assertion, you may just find the linear recurrence of A006139, and iterate to high $n$, say 5000,

PiRec = (1 + n) a[n] + 1/2 (-3 - 2 n) a[1 + n] + 1/4 (-2 - n) a[2 + n];
BigData[Rec_, IC_, nMax_] := Transpose[{ 
RecurrenceTable[{Rec == 0, a[0] == 0, a[1] == IC[[1]]},  a, {n, 1, nMax}],
RecurrenceTable[{Rec == 0, a[0] == 1, a[1] == IC[[2]]},  a, {n, 1, nMax}]}]
Divide @@ # & /@ BigData[PiRec, {8, -2}, 6]
BigFrac=Divide@@BigData[PiRec, {8, -2}, 5000][[-1]]

Out[]:={-4, -3, -19/6, -160/51, -1744/555, -644/205}
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8462533777303074318457713307784901098342773205427549601914627415625)

The first problem is that we don't know if Mathematica calculates correctly to this order of magnitude. A check is given by the quality,

$MaxExtraPrecision = 10000;
{Denominator[BigFrac]^.79 N[Pi + BigFrac, 1000] > 1,Denominator[BigFrac]^.8 N[Pi + BigFrac, 1000] > 1}
Out[]:={False,True}

This measurement places $0.79<Q<0.80$, which seems like good news but may also contradict Beukers. Who or what has gone wrong in this situation? Is $n=5000$ not sufficiently far in the limit? Is it a fault of my programming? Of Mathematica? Or have we found a mistake in the original article? Expert opinions welcome.

I would avoid floating point exponents like .97 or .8 and do as much of the calculation with exact numbers as possible. If Mathematica is not wrong, Q seems to be .7934:

$MaxExtraPrecision = 10000;
{ Denominator[BigFrac]^(7934/10000) (Pi + BigFrac) > 1,
 Denominator[BigFrac]^(7945/10000) ( Pi + BigFrac) > 1}
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