I'm not a mathematician, so if this question is out of line, just ignore it. But I wonder at the use of a recursive computer program or infinite series to approximate some transcendental number, that uses, in the program, an irrational or even a transcendental function. Isn't the point of approximation programs to use only rational operations to achieve the approximation? Brent On 1/24/2020 8:19 PM, Brad Klee wrote:
For a horrible approximation of Pi/2, and a minor headache along the way:
Ann2 ={1, -4 x, 4 (1 - x) x}; ERev1 = Hypergeometric2F1[1/2, 3/2, 2, x] x; ERev2 = ERev1*Integrate[ Series[Exp[-Integrate[Ann2[[2]]/Ann2[[3]], x] ]/ERev1^2, {x, 0, 100}], x]; Limit[D[EllipticE[1 - x] - ERev2, x], x -> 0] N[Normal[ERev2 + %*ERev1 - Pi/2] /. x -> 1] Out[] = (Log[2] - 5/8) Out[] = -0.00155653
Where did this factor (Log[2]-5/8) come from? Is asymptotic analysis the only option we have left?
After a few hours today, I still have yet to come up with a convincing answer, and feel thoroughly defeated (by a computer program no less).
Am I missing something? Is this (Log[2]-5/8) somehow easy to derive?
--Brad
See also:
http://functions.wolfram.com/EllipticIntegrals/EllipticE/06/01/04/01/
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