  We write an action, in four dimensional N=1 curved superspace, which contains
a pure R^4 term with a coupling constant. Starting from the off-shell solution
of the Bianchi identities, we compute the on-shell torsions and curvatures with
this term. We show that their complete solution includes, for some of them, an
infinite series in the R^4 coupling constant, which can only be computed
iteratively. We explicitly compute the superspace torsions and curvatures up to
second order in this coupling constant. Finally, we comment on the lifting of
this result to higher dimensions.
