quantum mechanics - Relation between perturbation theory and Taylor expansion in QM

So I am looking at non-degenerate perturbation theory. The idea is that the perturbing term in the Hamiltonian is small so you somehow expand the energies and wave functions in this small term and collect orders. Now I did an exercise in which you apply perturbation theory to a system, which is solvable. You then show by Taylor expanding the analytical result of the energies that the first order perturbation term is equal to the first order term in the Taylor expansion. Should this be obvious? I know that the first order perturbation theory was derived based on expanding the energies in the small perturbing term but somehow I cannot see that it is exactly equivalent to simply calculating the first order term in the energy.