The Lagrange inversion theorem in the smooth case

The classical Lagrange inversion theorem is a concrete, explicit form of the implicit function theorem for real analytic functions. An explicit construction shows that the formula is not true for all merely smooth functions. The authors modify the Lagrange formula by replacing the smooth function by its Maclaurin polynomials. The resulting modified Lagrange series is, in analogy to the Maclaurin polynomials, an approximation to the solution function accurate to o(xN) as x→0.