  We obtain a new decomposition of the Riemann-Liouville operators of
fractional integration as a series involving derivatives (of integer order).
The new formulas are valid for functions of class $C^n$, $n \in \mathbb{N}$,
and allow us to develop suitable numerical approximations with known
estimations for the error. The usefulness of the obtained results, in solving
fractional integral equations and fractional problems of the calculus of
variations, is illustrated.
