  A recently derived method [R. D. Rohrmann and A. Santos, Phys. Rev. E. {\bf
76}, 051202 (2007)] to obtain the exact solution of the Percus-Yevick equation
for a fluid of hard spheres in (odd) $d$ dimensions is used to investigate the
convergence properties of the resulting virial series. This is done both for
the virial and compressibility routes, in which the virial coefficients $B_j$
are expressed in terms of the solution of a set of $(d-1)/2$ coupled algebraic
equations which become nonlinear for $d \geq 5$. Results have been derived up
to $d=13$. A confirmation of the alternating character of the series for $d\geq
5$, due to the existence of a branch point on the negative real axis, is found
and the radius of convergence is explicitly determined for each dimension. The
resulting scaled density per dimension $2 \eta^{1/d}$, where $\eta$ is the
packing fraction, is wholly consistent with the limiting value of 1 for $d \to
\infty$. Finally, the values for $B_j$ predicted by the virial and
compressibility routes in the Percus-Yevick approximation are compared with the
known exact values [N. Clisby and B. M. McCoy, J. Stat. Phys. {\bf 122}, 15
(2006)]
