paradox, accurate taylor series half iterate of eta not analytic at e

[sheldonison]

Hey Sheldon, its not that paradoxical.
These series are called asymptotic powerseries.
.....
In our case the half-iterate has this kind of asymptotic development.
So if we glue the left and right half-iterate together at e,
then all derivatives exists and are finite at e, but still the function is not analytic at e, i.e. the convergence radius is 0.
....

Thanks for the information! So does this mean, the infinite series will diverge, no matter how small abs(z-e) is? But the truncated finite series may be fairly accurate, depending on how many terms of the series are included, and abs(z-e)?

I'm convinced that the series terms, \( a_n(z-e)^n \), start getting larger for terms beyond a40 or so, and the series begins to diverge for |z-e|=1.
- Sheldon