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The quasilinearization method is traditionally developed with quadratic convergence. In this paper the method is extended to higher order convergence for nonlinear Riemann-Liouville fractional differential equations of order 0<q<1. Existence and comparison results of the linear RL fractional differential equations are recalled and modified where necessary. Using upper and lower solutions, sequences are constructed that are monotonic such that the weighted sequences converge uniformly, with order higher than 2, to the unique solution of the differential equation. A numerical example illustrating the main result is given.