  The effective-action formalism is applied to a gas of bosons. The equations
describing the condensate and the excitations are obtained using the loop
expansion for the effective action. For a homogeneous gas the Beliaev expansion
in terms of the diluteness parameter is identified in terms of the loop
expansion. The loop expansion and the limits of validity of the well-known
Bogoliubov and Popov equations are examined analytically for a homogeneous
dilute Bose gas and numerically for a gas trapped in a harmonic-oscillator
potential. The expansion to one-loop order, and hence the Bogoliubov equation,
is shown to be valid for the zero-temperature trapped gas as long as the
characteristic length of the trapping potential exceeds the s-wave scattering
length.
