  We investigate the sedimentation of identical inertialess spherical particles
in a Stokes fluid in the limit of many small particles. It is known that the
presence of the particles leads to an increase of the effective viscosity of
the suspension. By Einstein's formula this effect is of the order of the
particle volume fraction $\phi$. The disturbance of the fluid flow responsible
for this increase of viscosity is very singular (like $|x|^{-2}$).
Nevertheless, for well-prepared initial configurations and $\phi\to 0$, we show
that the microscopic dynamics is approximated to order $\phi^2 |\log \phi|$ by
a macroscopic coupled transport-Stokes system with an effective viscosity
according to Einstein's formula. We provide quantitative estimates both for
convergence of the densities in the $p$-Wasserstein distance for all $p$ and
for the fluid velocity in Lebesgue spaces in terms of the $p$-Wasserstein
distance of the initial data. Our proof is based on approximations through the
method of reflections and on a generalization of a classical result on
convergence to mean-field limits in the infinite Wasserstein metric by Hauray.
