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

[bo198214]

So does this mean, the infinite series will diverge, no matter how small abs(z-e) is?

Yes.

But the truncated finite series may be fairly accurate, depending on how many terms of the series are included, and abs(z-e)?

Yes.

But there is a trick you can use. Take a truncated powerseries of at least degree 2 (or in general case \( f(x)=a+x+c(x-a)^{m+1}+\dots \) of degree \( m+1 \)) of the intended by \( t \) iterated powerseries, i.e. a polynomial \( \pi \), of the form \( \pi(x)=a+x+tc(x-a)^{m+1}+\dots \).

Then one knows that
\( f^{\pm n}(\pi(f^{ \mp n}(z))) \)
converges to the precise value of the fractional iteration (for \( n\to\infty \)), where you choose the opposing signs so that the inner \( f^{\mp n}(z) \) converges towards the fixpoint \( a \) (and the outer \( f^{\pm n} \) then takes it back).

Its because the series gets more precise - so to say - the closer the value is at the fixpoint a. And because \( f^{\pm n}\circ f^t \circ f^{\mp n}=f^t \).