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/