I would like to understand better the closure properties of the operation of taking the Taylor expansion. Let $f(x) = h(g(x))$.
I believe that the composition of the Taylor expansions of $g$ and $h$ works as a polynomial expansion of $f$ only if the codomain of the function $g(x)$ and its Taylor expansion, falls within the radius of convergence of the Taylor expansion of the function $h$.
Example: If I want to find the polynomial expansions of $\sqrt{-log(x)}$, I can NOT compose the Taylor expansion of $\log(x)$ in $1$ and the Taylor expansion of $\sqrt{x}$. This because the codomain of $-\log(x)$ for (say) $x=0.0001$ is $9.210$ which falls away of the ratio of convergence of $\sqrt{x}$ expanded in (say) $1$.
Am I right?
Where can I find a proof that I can compose only polynomial expansions of functions where the radius of convergence is preserved?
There are no guarantees about the radius of convergence of compositions.
For example, if $g(x) = \frac 1{1 - x}$, its Taylor series about $0$ has radius of convergence $1$. In particular $1$ is not in its domain. Similarly $h(x) = \frac 1x$ has a Taylor series of radius $1$ about $1$. But $f=h(g(x))$ has a Taylor series about any point you choose with an infinite radius of convergence. Indeed all but the first two coefficients will be $0$.