Our main result is that it is possible to express the Schwinger functions (or the effective potentials) as formal power series of objects which we call ''form factors'' which, although divergent to all orders of perturbation theory if the cut-off N is removed, obey to all orders a formal equation which retains its meaning as \(N\to \infty\). We show that if the formal equation admits a solution verifying suitable bounds, then the formal power series for the Schwinger functions in terms of the form factors is bounded to all orders. Hence there is the possibility of giving a meaning to perturbation theory of nonrenormalizable interactions without introducing infinitely many new counterterms, but rather introducing infinitely many new constants, the form factors, which however are not independent but are related by an equation.