into a non-regular continued fraction. E.g., threewise grouping the continued fraction for e, 1,0,1,1,2,1,1,4,1,1,6,1 ... gives the general matrix product Product[{{2 + 2 n, 1 + 2 n}, {1 + 2 n, 2 n}}], {n, ∞}] which converts to the non-regular continued fraction 2 - ContinuedFractionK[(2 n - 3) (2 n + 1), 8 n^2 - 4, {n, ∞}] == E which must converge exactly thrice as fast as the regular one: In[82]:= ContinuedFraction[%[[1]] /. ∞ -> 9] Out[82]= {2, 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, 1, 1, 10, 1, 1, 12, 1, 1, 14, 1, 1, 16, 1, 1, 19} In[83]:= Length@% Out[83]= 27 (Note that 19 = 18+1/1 .) This suggests that non-regular continued fractions could be abandoned in favor of matrix products. (Which will require the cooperation of computer algebraists. E.g., Mathematica has no non-commutative Product (E.g., Dot[...,{n,1,∞}]) expression. It doesn't even have Table (List) over a symbolic range.) —rwg