Why do we use perturbative series if they don't converge? My course instructor mentioned that the Perturbative Series are not convergent but diverge as we consider more and more terms in the expansion. He then briefly mentioned that the Perturbative Series are Asymptotic Series. I have some idea about Asymptotic series, such as Stirling's approximation $\left (n!\sim {\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n} \right )$, which gets better as $n$ increases.
So, does it mean that the perturbative expansion of $n^{th}$ state energy ($E_n$) is gets better as $n$ increases?
 A: The difference between a convergent and asymptotic series comes from reversing the order of two quantifiers.

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*Convergent: For every $x$, there exists a large enough number of terms such that the error is less than $\epsilon$.

*Asymptotic: For every number of terms, there exists a small enough $x$ such that the error is less than $\epsilon$.

The utility of the latter can be summed up by Carrier's rule: "Divergent series converge faster than convergent series because they don't have to converge". In other words, the approximation will get better for awhile and then start to get worse at a point that depends on $x$.
In the case of Stirling's approximation, the small number $x$ would be $1/n$. For perturbation theory in quantum mechanics though, it is some coefficient of an interaction Hamiltonian. I'm not aware of any result stating that in systems with a discrete spectrum, excited state energies are more easily approximated than the ground state energy. The eigenstate thermalization hypothesis more or less tells us to expect the opposite.
