Infinite systems of equations appear naturally in combinatorial counting problems. Formally, we consider functional equations of the form &y;(x)=F(x,&y;(x)), where F(x,&y;):&C;×ℓp→ℓp is a positive and nonlinear function, and analyze the behavior of the solution &y;(x) at the boundary of the domain of convergence. In contrast to the finite dimensional case different types of singularities are possible. We show that if the Jacobian operator of the function F is compact, then the occurring singularities are of square root type, as it is in the finite dimensional setting. This leads to asymptotic expansions of the Taylor coefficients of &y;(x).