  Using a high temperature virial expansion, we present a controllable study of
the thermodynamics of strongly correlated Fermi gases near the BEC-BCS
crossover region. We propose a practical way to determine the expansion
coefficients for both harmonically trapped and homogeneous cases, and calculate
the third order coefficient $b_{3}(T)$ at finite temperatures $T$. At
resonance, a $T$-independent coefficient
$b_{3,\infty}^{\hom}\approx-0.29095295$ is determined in free space. These
results are compared with a recent thermodynamic measurement of $^{6}$Li atoms,
at temperatures below the degeneracy temperature, and with Monte Carlo
simulations.
