  Four point correlation functions for many electrons at finite temperature in
periodic lattice are analyzed by the perturbation theory with respect to the
coupling constant. The correlation functions are characterized as a limit of
finite dimensional Grassmann integrals. A lower bound on the radius of
convergence and an upper bound on the perturbation series are obtained. The
perturbation series up to second order is numerically implemented along with
the volume-independent upper bounds on the sum of the higher order terms in 2
dimensional case.
