Perturbation theory emitting high order powers For my second-order energy correction for a harmonic oscillator in an electric field I have the following:
$$q^2\varepsilon^2\sum_{m\neq n}\frac{|\langle m|x|n\rangle|^2}{E^{(0)}_n-E^{(0)}_m}+\text{ }...$$


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*My first question is what values of $m$ do we sum over? 

*Secondly, my textbook says we neglect higher-power terms, does this mean higher values of $m$ or what? 
If I have left any information out (which is very likely) just say.
 A: No it doesn't mean higher values of m, as m here represents the set of orthonormal eigenstates of the system under study. In order to better understand where and how one arrives at 1st, 2nd, 3rd, ... order corrections, you need to look at the derivation of time independent perturbation theory that you find in most QM textbooks, there's also a rather detailed derivation on wikipedia.
An overview:
When you write out the time-independent Schrödinger equation for the perturbed system: $$(H_0 + \lambda V)\left|n\right > = E_n \left|n\right > $$
Where $H_0$ is the Hamiltonian of the unperturbed system, $\left|n\right >$ are the eigenstates of the perturbed system, $\lambda$ is a perturbation parameter describing its strength, $E_n$ energy eigenvalues, and $V$ can be an external field seen as perturbative. Now for small perturbations, i.e. small $\lambda$, we can expand the above equation as a power series of $\lambda$:
$$E_n = E_n^{0} +\lambda E_n^{1}+\lambda^2 E_n^{2}+\lambda^3...$$
and $$\left|n\right >=\left|n^0\right >+\lambda\left|n^1\right >+\lambda^2\left|n^2\right >+...$$
Substituting the expanded version into Schrödinger's equation again, you get a system of equation with as many equations as the power of $\lambda$ you intend to keep, $\lambda^0,\lambda^1,\lambda^2$..., solving for each one of them brings you to:
First order correction (expectation value of the perturbation Hamiltonian taken in the unperturbed state): $$E_n^1=\left<n^0 \right|V\left|n^0\right >$$
Second order: $$E_n^2= \left<n^0 \right|V\left|n^1\right > = \sum_{m\ne n}\frac{\left|\left<m^0 \right|V\left|n^0\right >\right|^2}{E_n^0-E_m^0}$$
Third order: $$E_n^3= \left<n^0 \right|V\left|n^2\right >=... $$
All this only valid for non-degenerate states, as I didn't not take into account degeneracy.
EDIT: Concerning your ''m'' confusion: Example of m: Take the hydrogen atom, first eigenstate would be $(n,l,m_l)=(1,0,0)$, second state $(2,0,0)$ and so on... (remember the wave-functions...)
