Thank you, Geogre, You mean psi?

THANK YOU GEORGE! OK (je vais voir tout ça).

George, thank you again,

I could't find the full text of Duverney, The two other papers I found, read, excellent... You sent me to my previous knowledge in numerical methods, good refresh. Yes, they use the solution of the minimal polynomial of the fib series presented recursively, that seems to me normal as the difference between two consecutive members in the reciprocal psi - series tends to it (GR).

My imagination is too small however to see how precisely this is related to real processes, information, entropy, systems. How these faster convergent series can suggest me this? Becouse they are alternate perhaps? Physics, waves...

A colleague of mine suggested me this reading and I found it excellent:

http://www.kurims.kyoto-u.ac.jp/~kyodo/kokyuroku/contents/pdf/1852-18.pdf

The author, Mr Aurues Takashi, is on RG.

Hm. Something with the reciprocal fibs should be like that, no? Suggestions?

Thank you a lot, George. I will write you soon.

Dear Velina,

"how precisely is this related to real processes, information, entropy, systems. How these faster convergent series can suggest me this? Becouse they are alternate perhaps? Physics, waves...Why should the reciprocal Fibonacci appear in the approximation of a process? "

Although neither of the authors below appears specifically (to my limited awareness) to discuss Fibonacci manifestations  I have a strong gut feeling that the underpinning explanations are to be sought  in the perspectives of Synergetics, developed by Hermann Haken (from ideas inspired by laser theory, non-equilibrium thermodynamics and Schrödinger's "order based on order" proposal for the organization of living matter), and of Scott Kelso's Dynamic Patterns theory, which extends synergetics thinking to account for the brain's capacity to organize and co-ordinate goal-related motor behaviour.

For literature by Haken, Kelso (1995, op. cit. below) recommends:

The Science of Structure: Synergetics. New York:Van Nostrand Reinhold, 1984

Synergetics, an Introduction: Nonequilibrium Phase Transitions and Self-Organization in Physics, Chemistry, and Biology, 3rd rev. enl. ed. New York: Springer-Verlag, 1983.

Advanced Synergetics: Instability Hierarchies of Self-Organizing Systems and Devices. New York: Springer-Verlag, 1993 

For the contents of the last:http://tocs.ulb.tu-darmstadt.de/12242607X.pdf

For Kelso:

Kelso, J.A. Scott (1995): Dynamic Patterns: The self-organization of brain and behavior , MIT Press (Preview - http://www.amazon.co.uk/Dynamic-Patterns-Self-Organization-Behavior-Adaptive/dp/0262611317/ref=sr_1_1?s=books&ie=UTF8&qid=1454938777&sr=1-1&keywords=9780262611312

which I've been reading over recent months. Possibly you also might find the following chapter, co-authored by Kelso, stimulating, and perhaps offering some pointers:

Kugler, P. N., Kelso, J. S., & Turvey, M. T. (1982). On the control and coordination of naturally developing systems.  In: Kelso, J.A.S and Clark, J.E.(eds.)The development of movement control and coordination, 5, 78. John Wiley & Sons. 1982 http://www.haskins.yale.edu/Reprints/HL0378.pdf  

Warmest wishes, Richard.

Dear Richard,

This is GREAT! Thank you!

Wormest wishes,

Velina

I have studied the method to find out in few steps the approximation ratio for any square root by the sequence  seed-a, seed-b, ..., a, b, new-b =mb+na, new-a = b, Result is b/a*2-n = √(n²+4m).

√(x): n=4, m= 4x-4. You get simple ratio with integers. For example the sequence 1/2(a+x/a) ->a ->√x gives more complicated sum of ratios.