Remy Cano on Mon, 25 Jan 2016 01:00:12 +0100

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Golden phi itself...


Additionaly for my last e-mail about N viewed as the square of an infinite series...

Another direct consequence.

IF you pick phi = 1.618...  yes, the golden ratio number (the one with the plus sign)
take its reciprocal,  square it, and perform an infinite sum with its positive powers
starting from zero,

THEN what you get as result is phi itself!!!

Isn't it nice?.

At purpose here is an informal technical proof.
It was found thankful for the aid of PARI.


             GP/PARI CALCULATOR Version 2.7.5 (released)
      i686 running linux (ix86/GMP-5.0.5 kernel) 32-bit version
         compiled: Jan 24 2016, gcc version 4.7.2 (GCC)
                        threading engine: single
     (readline v5.2 enabled, extended help not enabled)

                Copyright (C) 2000-2015 The PARI Group

PARI/GP is free software, covered by the GNU General Public License, and comes WITHOUT ANY WARRANTY WHATSOEVER.


? \p 10000
   realprecision = 10008 significant digits (10000 digits displayed)                                    
? (sqrt(5)+1)/2 - suminf(k=0,((sqrt(5)+1)/2)^(-2*k))
time = 12min, 4,544 ms.                                                                                 
%10 = 1.086922717 E-10004