Higher-order (e.g. $n$th) corrections in quantum perturbation theory (non-degenerate, time-independent) In perturbation theory for quantum mechanics, using Schrodinger equation and the expansions $$H=H_0+\varepsilon H_1+\varepsilon^2 H_2+\cdots$$
and $$E_n=E_n^{(0)}+\varepsilon E_n^{(1)}+\varepsilon^2 E_n^{(2)}+\cdots$$
one can get the $1$st,$2$nd or even $3$rd,$4$th order corrections of eigenenergy $E_n$ and the corresponding wave function corrections.
(Non-degenerate, time-independent)
But does there exist systematic(?) series (not Dyson series) to attain higher order (6th,7th and more) corrections of them? (The method is clear, but the results are too complicated, are there any simplified results?)
 A: Dyson series are specifically geared for many body problems. However, if you have been that far, you are probably also familiar with the Brueckner-Goldstone expansion for the ground state energy that in principle is applicable for any type of system.
The perturbation theory that you cite is known as Rayleigh-Schrödinger perturbation theory - it is used for eigenvalue problems, where we try to determine the eigenstates and eigenenergies. There is its close relative Brilluoin-Wigner perturbation theory, which is used for dealing with scattering problems, and where the energy is usually given, which greatly simplifies obtaining general formulas. Dyson expansion is essentially a more sophisticated form of this perturbation theory - even when applied in condensed matter. One then uses some additional relations to determine the ground state energy, thermodynamic potentials, etc. - these can be again teased out from the many-body texts.
Caveat
Nice closed form expressions for infinite series do not necessarily make life easier. E.g., using Goldstone theorem to rederive ground state of an exactly solvable system can prove to be quite a pain.
