Passive scalar convection by a prescribed random velocity field is represented in terms of integral equations. Primitive perturbation expansions are constructed by iterating these integral equation representations as in Kraichnan (1977). First and second iterations of elemental functions within these expansions are assumed quadratically integrable with respect to space and time. That is, they are assumed to belong to the space L2. Line-renormalized perturbation expansions are constructed, corresponding to these primitive perturbation expansions, which converge almost everywhere. The direct-interaction approximation and the Lagrangian-history direct-interaction approximation are the simplest truncations of the appropriate line-renormalized perturbation expansions.