We present a series expansion calculation of the bulk effective coefficient of weakly nonlinear behavior in some continuum composite conductors and in simple cubic random resistor networks. The expansion is in powers of the relative difference between the linear Ohmic conductivities of the components. It is carried up to third order for an independent random bond network and a diagrammatic scheme is used to aid in implementing the calculation. For continuum composites, only the first term of the expansion can be calculated explicitly without detailed information about the microgeometry. Such information is difficult to acquire and even more difficult to exploit.