Conference rusure::math

690.0. "Recurrence relation inquiry from USENET" by SQM::HALLYB (Are all the good ones taken?) Thu Apr 09 1987 11:29

Path: decwrl!decvax!tektronix!uw-beaver!teknowledge-vaxc!sri-unix!husc6!seismo!mcvax!inria!imag!phs
Subject: Solution of a Recurrence Relation
Organization: IMAG, University of Grenoble, France
 
 
   In the solution of a puzzle, I ended up with the recurrence relation :
 
                    1                              1   2
	      X  = ---               X    = X  -  --- X
               1    2                 n+1    n     2   n
 
 
   Obviously X is decreasing, positive and has 0 as a  limit. Even more  it
is roughly equal to 2/n,  but in order to get  a more precise  idea of what
could be the value of X for "large" n's, I tried to express it as a sum :
             
                     a1     a2     a3
         X  =  a0 + ---- + ---- + ---- + ...
          n          n      n^2    n^3
 
but found no solution.
 
   Is  there someone who could give  me an equivalent of  X  for large  n ?
With explanations ? I am  surprised that this function  has no  development
because it is   such a "simple  and smooth  function";  is there  a general
theory of which functions can be developed, and under which forms ?
 
Thanks in advance.
--
Philippe SCHNOEBELEN,
LIFIA - IMAG,  BP 68                          UUCP : ...mcvax!imag!lifia!phs
38402 Saint Martin d'Heres, FRANCE
 
"Algebraic symbols are used when you do not know what you are talking about."

    The latest approximation gives something like 2/(n+log(n)+O(1)).
    (that's a natural logarithm, naturally).