Suppose we have to functions $g(x)$ and $f(x)$ that are polynomials of the same degree that is \begin{align*} g(x)=\sum_{k=1}^n a_k x^k\\ f(x)=\sum_{k=1}^n b_k x^k \end{align*}
then we have a following statment:
If $ |a_n| \ge |b_n|$ than there exists $x_0$ such that $|g(x)| \ge |f(x)|$ for all $x \ge x_0$.
My questions: Now suppose that $g(x)$ and $f(x)$ are power series, can we have a similar statement. For example, if \begin{align} \lim_{n \to \infty} \frac{|a_n|}{|b_n|} \ge 1 \end{align} then eventually $|g(x)| \ge |f(x)|$.
If $\lim \inf a_n - b_n > 0$, that means there exists $N$ so $n>N \Rightarrow a_n > b_n$, then $f(x) > g(x)$ eventually for increasing $x$. Just find The suitable $x_0$ (if < 0 then use 0) for the truncation at $N$ and then note that all following terms of the $f$ expansion are greater than the corresponding ones in the $g$ expansion