  A detailed study of the rescattering series is performed within a model,
using a generalized procedure of eikonalization and fitted to the $pp$ and
$\bar pp$ elastic scattering data. We estimate and compare the various
rescattering corrections to be added to the Born contribution in the amplitude.
We find that their number is finite, whereas it increases with the energy and
the transfer, like does their importance. In the domain where data exist, we
find also that a correct computation must include, at least, all two- and
three-Reggeon exchanges and some four- and five-Reggeon exchanges. Any
approximation aiming to reduce this (large) number of exchange would be
hazardous, especially when extrapolating. We extend our estimates in the domain
of future experiments.
