  Perturbative expansions, the Born series, of the scattering length and the
amplitude of $S$-wave neutron-deuteron scattering for spin quartet channel
below deuteron breakup threshold are studied in pionless effective field theory
at small cutoff values. A three-body contact interaction is introduced when the
integral equation is solved using the small cutoffs. After renormalizing the
three-body interaction by using the scattering length, we expand the integral
equation as the ordinary and inverted Born series. We find that the scattering
length and the phase shift are considerably well reproduced with a few terms of
the inverted Born series at relatively large cutoff values, $\Lambda\simeq 100$
MeV.
