Given an analytic function $f: \mathbb{R} \to \mathbb{R}$ whose Taylor series converges over all $\mathbb{R}$ and is
\begin{equation} f(z) = \sum_{k=0}^{\infty}a_k x^k, \end{equation} and where the coefficients $a_k$ are all that is known (i.e. the closed form of the function is unknown), is it possible to extract somehow the behaviour of the function for $x \to \infty$ (assuming it exists)? That is, is it possible to find $f(\infty)$ from $\{ a_k \}$?
In principle the coefficients $a_k$ uniquely determine the function $f$ and therefore its behaviour at $\infty$. In practice, not much can be said in general. Well, here's something not quite general, but maybe better than nothing.
Suppose $f$ is analytic in the set $\{z \in \mathbb C: |1 - e^{-z}| < 1\}$ (which includes the positive real axis). Letting $w = 1 - e^{-z}$, we have $z = -\log(1-w)$. Then the function $f(z) = f(-\log(1-w)) = g(w)$ can also be written as $g(w) = \sum_{j=0}^\infty b_j w^j$ where each $b_j$ can be determined from $a_0, a_1, \ldots, a_j$ using Faa di Bruno's formula: $$\eqalign{b_0 &= a_0\cr b_1 &= a_1\cr b_{{2}}&=\dfrac{1}{2}\;a_{{1}}+a_{{2}}\cr b_{{3}}&=\dfrac{1}{3}\;a_{{1}}+a_{{2}}+a_{{3}}\cr b_{{4}}&=\dfrac{1}{4}\;a_{{1}}+{\dfrac {11}{12}}\; a_2 +\dfrac{3}{2}\,a_{{3}}+a_{{4}}\cr \ldots}$$ If $\sum_{j=0}^\infty b_j$ converges, it converges (by Abel's theorem) to $\lim_{w \to 1-} g(w) = \lim_{z \to +\infty }f(z)$.