Behaviour of function defined by Taylor series

I am trying to understand if the knowledge of the Taylor coefficients of a function can be used to understand the behaviour of the function.

For example, let $f \colon \mathbb R \to \mathbb R$ be the analytic function defined by $$ f (x) = \sum_{k = 1}^\infty \frac{x^k}{2^{k^3}} $$ How can one visualize this function, for example its behaviour at infinity? Can one give a formula in terms of (integrals of) elementary functions? (For some reason, I can't get WolframAlpha to plot this.)

Is there a function roughly of this form (whose Taylor coefficients are bounded by $C^{-k^2}$ for some constant $C > 0$) which tends to $0$ as $x \to \pm \infty$?

The answer to your question is no (assuming $C>1$ as for $C \le 1$ the answer is trivially yes as $e^{-x^2}$ shows) as any entire (nonconstant) function of order less than $1/2$ cannot be bounded on any half line (it follows from Lindelof applied to $f(z^2)$)

Since for $f(z)=\sum a_nz^n$ the order is given by $\limsup \frac{n \log n}{-\log |a_n|}$ this means that we cannot have $\frac{n \log n}{-\log |a_n|} <1/2-\epsilon$ for $n$ large enough or equivalently we cannot have $|a_n| << e^{-(2+\epsilon)n\log n}$ for any $\epsilon>0$ (and of course $e$ can be replaced by any $C>1$)

The result is sharp as $\cos \sqrt z=\sum \frac{(-1)^nz^n}{(2n)!}$ shows