logit coefficients growth pattern

[bo198214]

The second reference is a holy grail. It explicitly proves the above expansion, and proves that the coefficients \(b_k\) are of "Gevrey class 1"--which is their way of saying the above bound. Which is the equivalent of saying that it is Borel summable.

Yeah, that's the thing ... I recently always had a tab open in my browser with this MO article
Does the formal power series solution to f(f(x))=sin(x) converge?

Actually I went there by incident. Then and when I was musing about was written there.
Will Jagy directly asked Ecalle and he told him that it is Gevrey class 1/p (where p - Ecalle calls it valit(f) - is the index up to which the coefficients of f equal the ones of the identity function).
But I didn't make the connection the growth of the coefficient - because I simply didn't know&read about Gevrey classes!

Great dig, James!

So is there something like a radius of convergence for Borel-summations? How far from the fixed point would it converge/give correct values?
And then I think it would give the correct solution for each petal, right James?

EDIT: It would be really interesting where the break between the petals is when one does the Borel summation ...