Is there a characterization of the sequences of non-negative real numbers $\{a_n\}_{n=0}^{\infty}$ such that $$ f(x)=\sum_{n\geq 0}a_n (-x)^n $$ converges absolutely for all $x\geq 0$?
It's not hard to see that this condition fails to hold whenever $a_n$ decreases at most exponentially in $n$. When $a_n$ decreases superexponentially in $n$, things get trickier; the only examples I can think of are based on $e^{-x}$, and they satisfy the condition.
More quantitatively: Is it true that the above condition holds if and only if $\limsup_n \frac{1}{n}\log a_n=-\infty$?