Suppose that we are given two real valued, continuous, infinitely differentiable functions $f_1$ and $f_2$ represented by their Taylor series around $x_0$ with coefficients $c_{1,i}$ and $c_{2,i}$, respectively. The radii of convergence satisfy $R_1\ge R_2$, while for every pair of coefficients we may say that $c_{1,i}\le c_{2,i}$.
It's intuitive that within the smaller radius $R_2$, where both series converge, the functions compare as $f_1\le f_2$. Can we generalize this conclusion (based on solely the pairwise coefficient comparison) to the region "above" the convergence radius $x\ge x_0+R_2$?
Edit: Suppose that all coefficients are positive (non-alternating series) and restrain the domain of interest to $x\ge x_0$.