Perturbative expansion of energy eigenstates If we add quartic term in quantum harmonic oscillator,
$$V(x)=\frac{mx^2}{2}+\frac{m^{2}\omega^{3}}{\hbar}\hat{x}^{4}.$$
$$H(\lambda)\,=\,H^{(0)}+\lambda\,\frac{m^{2}\omega^{3}}{\hbar}\dot{x}^{4}\,=\,H^{(0)}+\lambda\,\frac{1}{4}\,\hbar\omega(\dot{a}+\hat{a}^{\dagger})^{4}.$$
Using time independent perturbation theory, $$E_{0}(\lambda)=\frac{_1}{^2}\hbar\omega\left(1+\frac{^2}{2}\lambda-\frac{21}{4}\lambda^{2}+\frac{33}{8}\lambda^{3}-\frac{3585}{64}\lambda^{4}+\frac{916731}{128}\lambda^{5}-\frac{65518401}{812}\lambda^{6}+O(\lambda^{7})\right).$$
In MIT quantum Physics III course notes it is written that

As it turns out the coefficients keep growing and the series does not converge for any nonzero
$\lambda$; the radius of convergence is actually zero! This does not mean the series is not useful.
It is an asymptotic expansion. This means that for a given small value of $\lambda$ the magnitude
of successive terms generally decrease until, at some point, they start growing again. A
good approximation to the desired answer is obtained by including only the part of the sum
where the terms are decreasing.

I am not able to understand that how perturbative expansion is useful there. It says that for any value of $\lambda$, after some point the successive terms starts increasing. Then how the perturbative solution restricted to some order is useful. The main aim of perturbative expansion is that if $\lambda$ is small then the successive terms in the series starts decreasing and we can neglect them and the solution up to first or second order is a good approximate solution.
But how restricting the number of terms to the point where the terms starts increasing gives us the good approximate solution here as written in the notes? How can we neglect the higher order terms then? I think I am missing something.
Please explain.
 A: What we are dealing with here is asymptotic expansion:

In mathematics, an asymptotic expansion, asymptotic series or Poincaré expansion (after Henri Poincaré) is a formal series of functions which has the property that truncating the series after a finite number of terms provides an approximation to a given function as the argument of the function tends towards a particular, often infinite, point. Investigations by Dingle (1973) revealed that the divergent part of an asymptotic expansion is latently meaningful, i.e. contains information about the exact value of the expanded function.


Since a convergent Taylor series fits the definition of asymptotic expansion as well, the phrase "asymptotic series" usually implies a non-convergent series. Despite non-convergence, the asymptotic expansion is useful when truncated to a finite number of terms. The approximation may provide benefits by being more mathematically tractable than the function being expanded, or by an increase in the speed of computation of the expanded function. Typically, the best approximation is given when the series is truncated at the smallest term. This way of optimally truncating an asymptotic expansion is known as superasymptotics.1 The error is then typically of the form ~ exp(−c/ε) where ε is the expansion parameter. The error is thus beyond all orders in the expansion parameter. It is possible to improve on the superasymptotic error, e.g. by employing resummation methods such as Borel resummation to the divergent tail. Such methods are often referred to as hyperasymptotic approximations.

(Emphasis is mine.)
