On the asymptotic behaviour of continued fractions (Q1112267)

Let \(C_ n\) be the convergents of the continued fraction \[ b_ 0+\frac{a_ 1}{1}+\frac{a_ 2}{1-a_ 2}+\frac{a_ 3}{1-a_ 3}+.... \] Then the right-hand side of \(a_ n=-\Delta C_{n-1}/\Delta C_{n- 2}\) has a limit iff the continued fraction is limit periodic. The corresponding identity for suitable contractions of the continued fraction can be used for limit k-periodic continued fractions. Convergence acceleration using both familiar techniques for sequences and modifications of the continued fraction are discussed.