  We derive modular anomaly equations from the Seiberg-Witten-Donagi curves for
softly broken N=4 SU(n) gauge theories. From these equations we can derive
recursion relations for the pre-potential in powers of m^2, where m is the mass
of the adjoint hypermultiplet. Given the perturbative contribution of the
pre-potential and the presence of ``gaps'' we can easily generate the m^2
expansion in terms of polynomials of Eisenstein series, at least for relatively
low rank groups. This enables us to determine efficiently the instanton
expansion up to fairly high order for these gauge groups, e. g. eighth order
for SU(3). We find that after taking a derivative, the instanton expansion of
the pre-potential has integer coefficients. We also postulate the form of the
modular anomaly equations, the recursion relations and the form of the
instanton expansions for the SO(2n) and E_n gauge groups, even though the
corresponding Seiberg-Witten-Donagi curves are unknown at this time.
