See attached file.
I'll explain this in the Dirac notation, see attached pdf file for explanation of this notation.
Let's denote the unperturbed Hamiltonian by H . The eigenstates of H are the states |psi_n>, the corresponding eigenvalues are E_n:
H |psi_n> = E_n |psi_n>
Suppose that we perturb H. The Hamiltonian is now H' = H + X. A convenient way to find the corrections to E_n is to consider instead the Hamiltonian H'(epsilon) = H + epsilon X, work things out order by order in powers of epsilon, and then put epsilon = 1 at the end of the ...
A detailed explanation is given.