  The weak-coupling expansion for thermodynamic quantities in thermal field
theories is poorly convergent unless the coupling constant is tiny. We discuss
the calculation of the free energy for a hot gas of electrons and photons to
three-loop order using hard-thermal-loop perturbation theory (HTLpt). We show
that the hard-thermal-loop perturbation reorganization improves the convergence
of the successive approximations to the QED free energy at large coupling, e ~
2. The reorganization is gauge invariant by construction, and due to the
cancellations among various contributions, we obtain a completely analytic
result for the resummed thermodynamic potential at three loops.
