Let $f(x)$ be a real-valued function defined in $(0, \infty)$. I am curious what kind of $f(x)$ has the following representations: $$ f(x) = \sum_{j=0}^\infty a_j e^{-jx}, \quad \forall x \in (0, \infty). $$

My initial thought was that consider the transformation $s = e^{-x}, s \in (0, 1)$, then we want to find $\{a_j\}_{j=1}^\infty$ such that $$ f(-\log s) = \sum_{j=0}^\infty a_j s^j, \quad \forall s \in (0, 1) $$

That means we may consider the Taylor expansion for $f(-\log s)$ at $s=0$, but then $-\log s = \infty$, for which the Taylor series is not well defined (at infinity).

So is there a method to determine $a_j$ if $f(x)$ can be represented as an exponential sum?