[math-fun] dilog(1/2), log(2)

14 Nov 2005

      Hi,

this is my first message on math-fun. The following thing is very easy
to prove, but quite nice to look at:

when computing the product (which in fact is a continued fraction),

[0 4]                [x^4/n^2    2^n*(1-(n*x)-(n*x)^2+(n*x)^5]
[1 0] . prod(n=1,k,  [1          0                           ] )

then the quotient of the two terms in the left column (after an even
number of multiplications) or in the right column (after an odd number
of multiplications) converges to:

  dilog(1/2) x^4  +  log(2) x^5  +  O(x^6)

Here is the Pari/GP code:

  m=matrix(2,2,i,j,(i!=j)*j^2+O(x^6));
  for(n=1,100,
    m0=matrix(2,2);
    m0[2,1]=1;
    m0[1,1]=x^4/n^2;
    m0[1,2]=2^n*(1-(n*x)-(n*x)^2+(n*x)^5);
    m=m*m0);
  m[1,1]/m[2,1]

Regards,

-- 
Thomas Baruchel
  Home Page: http://baruchel.free.fr/~thomas/