comparing sequences via generating functions

Question

Suppose that we have two sequences of positive real numbers $\{ a_n \}$ and $\{ b_n \}$, and let $\displaystyle A(x) = \sum_{n=1}^\infty a_n x^n$ and $\displaystyle B(x) = \sum_{n=1}^\infty b_n x^n$ be their corresponding generating functions. If $a_n > b_n$ for all $n$, what can we say about $A(x)$ and $B(x)$? Is there also a converse property?

Answer 1

If $\forall n : a_n > b_n$ then $\forall x > 0$ such that $B(x) < \infty$, $A(x) > B(x)$ and moreover, for all positive $k$, $\frac{d^nA(x)}{dx^n} > \frac{d^nB(x)}{dx^n}$ wherever those derivatives are finite.

One possible converse would be that if $\forall x > 0 : A(x) > B(x)$ then $\forall n : a_n > b_n$, but that statement is false.

A converse which I believe is true is that if $\forall x > 0 : A(x) > B(x)$ and for all positive $k$ $\frac{d^nA(x)}{dx^n} > \frac{d^nB(x)}{dx^n}$, then $\forall n : a_n > b_n$.