Is it possible that a series of Feynman diagrams converge? A bit of maybe unnecessary context
I'm reading "Lecture notes on Diagrammatic Monte Carlo for the Frohlich polaron". It says 

It is usually unknown whether a series converges or not. The series is guaranteed to diverge at a phase transition, but it may happen sooner. In fact, most series in physics are asymptotic, which can be established rigorously in a number of cases

Question:
I take this as an indication that a series of Feynman diagrams may converge. However, I can't really make sense of it. To me it seems that no matter the system considered, each diagram will consist of a small parameter $u^N$. This parameter suppresses the importance of each diagram exponentially. However, for every(?) diagrammatic series the number of diagrams increases factorially. It now seems to me that for any finite $u$ the series will diverge, because the factorial number of diagrams always "beats" the expontential supressing.
I am not really sure how to understand this, but suspect I might have a wrong understanding of what is meant by convergence it this case.
In addition, I am aware of Dyson's argument that when the series is not analytic for a coupling constant equal to zero, the series will diverge. Hence, this question is only relevant when Dyson's argument does not apply.
 A: These series are power series like $\sum_{n=0}^{\infty}a_n g^n$ in some coupling $g$. Power series are special, versus all series in that one has a priori a good understanding of what can possibly happen. Namely, the radius of convergence $R$ defined by
$$
R=\frac{1}{\limsup\limits_{n\rightarrow \infty} |a_n|^{\frac{1}{n}}}
$$
with the convention that $1/0=\infty$ and $1/\infty=0$, is such that the following holds.

*

*If $|g|>R$ the series is trivially divergent, i.e., the general term $a_n g^n$ does not converge to zero.

*If $|g|<R$ the series converges and does so absolutely, i.e., $\sum_{n=0}^{\infty}|a_n g^n|<\infty$.

Now for typical Feynman diagram power series, $a_n=\sum_D b_D$ is a finite sum over diagrams $D$. Figuring out the size $|a_n|$ is more complicated than just counting how many $D$'s there are at a certain order $n$ in perturbation theory. This number is typically factorial, but there can be cancellations so that $|\sum_D b_D|$ ends up being much smaller than $\sum_D |b_D|$. Moreover, the contributions of the diagrams do not have the same size.
Now for a Bosonic theory like Euclidean $\phi^4$ with cutoffs, at fixed $n$, the $b_D$ are real numbers and they all have the same sign, so there cannot be any cancellations. If one neglects the variation of size of diagram contributions, one gets a rough estimate
$$
|a_n|\sim \frac{1}{n!}\times \frac{(4n)!}{2^{2n}(2n)!}\sim n!
$$
ignoring anything of the form $C^n$.
This results in $R=0$ and divergence of the series, no matter how small the coupling $g$ is.
For Fermionic theories, there are cancellations. In the fact, in the presence of cutoffs (UV and IR), the series is convergent for $g$ small, i.e., $R>0$.
Another notable model where the perturbation series converges in a very subtle way is the dipole gas. See

*

*K. Gawedzski and A. Kupiainen, "Block spin renormalization group for dipole gas and $(\nabla\varphi)^4$" in Ann. Phys.

*K. Gawedzski and A. Kupiainen, "Lattice dipole gas and $(\nabla\varphi)^4$
models at long distances: decay of correlations and scaling limit" in CMP.

Also note that Dyson's argument is nowhere near being a proof, it is just some handwavy heuristic. Finally, to get a better feel for these convergence issues, and resummation techniques like Borel's method, it is good to consider the pedagogical example of QFT in zero dimension as explained in: Rivasseau, "Constructive Field Theory in Zero Dimension".
A: The discussion in question deals with resummation of the diagrammatic series for a partition function. If the Hamiltonian and the phase space are properly defined, the partition function is finite. The non-convergence problem here is expanding a function at the point of its non-analiticity (e.g., one cannot expand $\log x$ or $1/x$ around $x=0$), and we do not know in advance whether it is analytical or not, as a function of the small parameter. We certainly know that it is non-analytical in the vicinity of a phase transition.
