Remark 1

If $g$ is of order $< 1$ relative to $x$, the function $u(x) - u_n(x)$ tends to 0 with $x$ once $n$ is sufficiently large. Indeed, in the opposite case one would have $g{g''}^n \gg 1$ for every $n$, so $g$ would be of infinite order relative to $1/g'$; in other words, one would have $\log|g| \gg \log|g'|$, whence, on differentiating, $g'/g \gg g''/g'$. But if $g$ is of order $\mu < 1$ relative to $x$ one has $g'/g \sim g''/g'$ when $\mu = -\infty, \frac{g'}{g} \sim \frac{\mu}{\mu - 1} \frac{g''}{g'}$ when $\mu \neq 0$, and finally $g'/g \ll g''/g'$ when $\mu = 0$ (V, p. 251, n 3).

In contrast, if $g$ is of order 1 relative to $x$ one can have $g{g''}^n \gg 1$ for every integer $n > 0$, as the example $g(x) = x/\log x$ shows.