  We study the virial coefficients B_k of hard spheres in D dimensions by means
of Monte-Carlo integration. We find that B_5 is positive in all dimensions but
that B_6 is negative for all D >= 6. For 7<=k<=17 we compute sets of Ree-Hoover
diagrams and find that either for large D or large k the dominant diagrams are
"loose packed". We use these results to study the radius of convergence and the
validity of the many approximations used for the equations of state for hard
spheres.
