This is a follow-up to my previous question. The set-up for this question is somewhat different. For $n \geq 1,$ let $M_n = (n \log n)^n$ and set $M_0=1.$ Say that an infinitely differentiable real function $f$ belongs to the quasi-analytic class $C\{M_n\}$if there are constants $\beta_f,B_f$ depending only on $f,$ such that $$||D^n f||_\infty \leq \beta_f (B_f)^n M_n \qquad \forall n \geq 0,$$ where $||-||_\infty$ is the supremum norm on $\mathbb{R}.$
In the book "Numerical methods in the study of Critical phenomena", pg. 30-32 , I have been told that the function $$b(x)= \sum_{n=1}^\infty \exp(-2^{n}/n) \cos(2^nx)$$ lies in $C\{M_n\},$ and that it is nowhere analytic. There is a purported proof that $b \in C\{M_n\}$ in the book "Numerical methods in the study of critical phenomena" (which is reproduced in this question). However, I strongly suspect that there are typos in the proof, and that it might even be wrong in some places. This makes the supposed proof unreadable to me.
So my question is:
How do I show that $b(x) \in C\{M_n\}?$ Can the proof given in "Numerical methods in the study of Critical phenomena" be salvaged?