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Long time ago I came up with the primitive 2 decimal digits Pi approximation:

Pi ~= Sqrt[4 E - 1]

see

https://oeis.org/A135821

and formula (9) in

https://mathworld.wolfram.com/PiApproximations.html

I was thinking how to improve it and in trying so lately came up with the following recurrence:

RecurrenceTable[{u[n + 1] == (1 + 1/u[n])^(Sqrt[4 E - 1] + 1),
  u[0] == Sqrt[4 E - 1] + 1}, u, {n, 0, 35}]

It appears that the even and odd indexed terms of the rational numbers sequence A(n), generated by the above recurrence, are converging to some limit value ? when

$n-->infinity$

A(n)={4.14215, 2.44921, 4.12963, 2.4552, 4.11755, 2.46102,
  4.10589, 2.46668, 4.09462, 2.4722, 4.08372, 2.47757,
  4.07317, 2.4828, 4.06296, 2.4879, 4.05306, 2.49287,
  4.04347, 2.49772, 4.03416, 2.50246, 4.02513, 2.50708,
  4.01636, 2.5116, 4.00783, 2.51601, 3.99954, 2.52033,
  3.99148, 2.52455, 3.98364, 2.52868, 3.976, 2.53271,...}

but I am using free version of Wolfram Alpha and it only works for the first 36 terms.

Could Mathematica help to evaluate the converging limit of above recurrence?

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  • $\begingroup$ If it helps you any, at 512 terms the even numbered terms have decreased to 3.50502 and the odd numbered terms increased to 2.82783 and the change appears to be getting smaller with every iteration. If the upper and lower values did converge to the same value then perhaps FindRoot[u==(1+1/u)^(Sqrt[4 E-1]+1),{u,3}] which returns 3.14152 might tell you something BUT you need to think very carefully to try to make certain that I haven't made any terrible mistake in that leap. $\endgroup$
    – Bill
    Jul 6, 2023 at 5:40
  • $\begingroup$ @Bill - Thanks!!! $\endgroup$
    – Alex
    Jul 6, 2023 at 13:07
  • $\begingroup$ At n=768 it has 3.44858 and 2.87081. Is there any chance that you could carefully turn that into two recurrence relations, one of the even terms and one of the odd terms, each with appropriate starting value? I do not know if that can be done, but since the difference between adjacent even terms and the difference between adjacent odd terms are so much smaller than the difference between even and odd terms that might open up the possibility of finding two limits and thus whether they are the same or different. Please be very careful with this. Thank you. $\endgroup$
    – Bill
    Jul 6, 2023 at 15:31
  • $\begingroup$ @Bill - thanks and yes, you are right, - the sequence (as is) has two distinct limit points and it doesn't converge in total. I got this clarification from the reply in community.wolfram.com/groups/-/m/t/… where I also posted my question. $\endgroup$
    – Alex
    Jul 6, 2023 at 16:31
  • $\begingroup$ Using @Bill's observation, we can expand the series of (1+1/u)^(Sqrt[4 E-1]+1)-u at the original approximation (1+1/u)^(Sqrt[4 E-1]+1) to get a better analytical approximation of the root value (notice I also use a substitution u->1/u which made the series coefficients much cleaner looking): originalapprox = Sqrt[4 E - 1]; uShifted[u_] = (1 + 1/u)^(1 + Sqrt[-1 + 4 E]) - u; Series[uShifted[u] /. (u -> 1/u), {u, 1/originalapprox, 1}] // Normal; soln = Simplify[1/(First@SolveValues[% == 0, u, PositiveReals])] .... $\endgroup$
    – ydd
    Jul 7, 2023 at 16:54

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