We rederive the Coulomb expansion of the electron gas average energy at finite temperature, starting from scratch, i.e., using only the framework of the grand canonical ensemble and not the finite-T Green's function formalism. We recover the analytical expressions of the exchange and correlation energy in both the high-T and the T= 0 limits. We explicitly show the origin of the crossover of the correlation energy leading term from e^4 In e 2 at zero temperature to e^3 at finite T. We also discuss the relative importance of exchange and correlation in both limits.