On the convergence behavior of continued fractions with real elements

Walter Gautschi
1983 Mathematics of Computation  
We define the notion of transient (geometric) convergence rate for infinite series and continued fractions. For a class of continued fractions with real elements we prove a monotonicity property for such convergence rates which helps explain the effectiveness of certain continued fractions known to converge "only" sublinearly. This is illustrated in the case of Legendre's continued fraction for the incomplete gamma function.
doi:10.1090/s0025-5718-1983-0679450-2 fatcat:bkeq2dqasjgtpgoal7tawbk24a