(This must be in Wall or Perron.) The cf with nth tail a[n]+b[n]/(a[n+1]+b[n+1]/ is computed by n a[n] b[n] P[n] p[n] prod ( ) = ( ) 1 0 Q[n] q[n] so that P[n+1] p[n+1] P[n] p[n] a[n] b[n] ( ) = ( ) ( ) Q[n+1] q[n+1] Q[n] q[n] 1 0 and P[n]/Q[n] - p[n]/q[n] estimates the uncertainty so far. Therefore P[n+1]/Q[n+1] - p[n+1]/q[n+1] r[n] := ----------------------------- P[n]/Q[n] - p[n]/q[n] as a measure of convergence is analogous to the term ratio of an ordinary series. I.e., -lg|rn| bits/term. Eliminating P, Q, p, and q, we get the peculiar result r[n] = a[n]*ccf[n+1]/b[n]-1, where ccf[n] is the "complementary cf" := -a[n]+b[n]/(-a[n+1] + b[n+1]/... . Approximating r[n+1] by r[n], r[n] ~ (sqrt(1+4*c[n])-1)/2/c[n] - 1 where c[n]:= b[n]/a[n]/a[n+1]. For a[n] := a*n^k+..., b[n] := c*n^(2*k)+..., r[n] -> (|a|*sqrt(4*c+a^2)-a^2)/2/c - 1 for large n. E.g., for Apery's continued fraction, k=3, a=34,and c=-1, giving r[oo] = 577-408*sqrt(2) = (1+sqrt(2))^-8. Note that for c->0 (e.g. linear/linear when k=1), r[n]->0. This is analogous to the term ratio -> 0 for e=sum(1/n!), which says that taking this limit won't tell us how many terms we need. At this point I planned to use the above to determine the optimal y for the twin cf Gamma formula but, experimentally, y=z kicks butt, and I'm too lazy to prove why. --rwg