Accelerating convergence of limit periodic continued fractionsK(an/1)

SummaryIt is shown that the convergence of limit periodic continued fractionsK(an/1) with liman=a can be substantially accelerated by replacing the sequence of approximations {Sn(0)} by the sequence {Sn(x1)}, where $$x_1 = - 1/2 + \sqrt {1/4 + a} $$ . Specific estimates of the improvement are derived.