Sub-series of a perturbation series and summation of infinite diagrams In many-body perturbation theory like in, Altland and Simons: Condensed matter field theory, 2nd edition, we express a correlation function in terms in an perturbation series.  My understanding is that mathematically speaking, this series is an asymptotic series (pg. 194 of the book mentioned) and does not converge and that the $n$th partial sum (sum of terms of perturbation series up to some order) has an error that first decreases (up to some order ~ 1/(coupling constant)) and then increases.
So that means that the best approximation of the correlation function is when we add all terms (i.e all diagrams) of the series up to some maximum order. But I have seen that often methods like RPA or summation of ladder diagrams sum a particular subset of 1-particle irreducible diagrams (using Dyson’s equation). I've seen that the book gives physical reasons for adding these diagrams (high electron density limit: pg 216), and says that it is a better approximation to do so. Why is it that this is true? In other words, why is an infinite sub-series of a perturbation series a better approximation?
 A: I thought I would put what I understand here hopefully will be of some relevance to your question asked.
The book 'Many-Body Quantum Theory of Condensed Matter Physics' by Bruus and Flensberg may have a suitable discussion that may satisfy you answer through out chapter 13.
I believe the case of have irreducible diagrams is inherent to if you cut the diagram with a fermion line you do not get two diagrams you have already accounted for (this would  be reducible if this was true). This may be a case of a double counting as such.
This is of particular relevance to the self-energy of a system, this is present in the definition  which includes the sum of all irreducible diagrams in the Green's Function, this is turn means you do not get a larger self-energy than you would expect from the system of study. The RPA summation of the self-energy however has another constraint such as the order matching the divergence number of the diagram but that is a separate constraint to the diagrams to the reducible nature.
