N, suminf( ), and the Golden ratio with PARI

[Remy Cano <reemmmyyyyccccaaanno@gmail.com>]

Greetings,

I hope you find this tiny script (.txt attachment )
and its comments useful for your analysis and
tests.

Humbly, with my best wishes.

Remy Cano

P.S.: Any moral support here is welcome.

After all, we learn everyday a little bit more.

/* (PARI) R. J. Cano, Jan 23 2016 */

/* ...
   ... Possibly helpful script in order to
   ... figure out something new about suminf()
*/

Si(N)=(-1+2*suminf(k=0,((N+1-2*sqrt(N))/(N-1))^k))^2;

Sf(N,U=99)=(-1+2*sum(k=0,U,((N+1-2*sqrt(N))/(N-1))^k))^2;
/*
 * ... There U=99 is just an arbitrary choice!,
 * ... it does mean of course to sum only up to the first 100 terms
*/

/*    
   ... The formula was achieved by trying (theoretically) to pack N-spheres
   ... centered over the longest diagonal of an N-cubic box in a way such
   ... that all these objects are tangent among them.

   ... This calculation should converge to N, this is: Si(N)==Sf(N,)==N
   ... But by setting \p to an enough high value, you will notice that
   ... Si(N) actually doesn't reach such limit due something that have
   ... to do with suminf() and the way it works. Of course it is
   ... alternatively fixed by shrinking the precision.

   ... Such defining formula also allow us to write the golden ratio
   ... number "Phi", as the infinite sum for the terms of a particular
   ... Geometric progression: suminf(k=0,((3-sqrt(5))/2)^k);
*/

Attachment: Untitled.png
Description: PNG image