The ultimate sanity check

[MphLee]

Hi Daniel. When using mathmode inline DO NOT USE single dollar sign,

Code:

$...$

use instead slash+round bracket

Code:

\(...\)

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Let's see if I'm getting this correctly. Expressions for non-integers iterations \(f^t(x)\) for \(t\) not integer, involve powerseries... thus infinite amount of terms to be summed.
No formal, abstract proof of convergence is available. No formal abstract proof of the identity of the powerseries associated with \(f^af^b\) and the powerseries for \(f^{a+b}\) is available, thus the comparison of the the value of the two infinite expression are made via direct computation of the \(N\)-truncation of the sums...

So in alternative to direct formal proof of power-series identity you use an experimental approach looking for concordance of the result up to some big O?

I hope I'm on the right track... I need to study a lot more but...
Trying to read Gottfried... doesn't the problem lies into going from finite approximations to infinite "divergent summation techniques"? Maybe is there where we lack of formal proofs that those semi-group identities holds?