Stieltjes transform and distributional solution

Question

let be the Stieltjes transform

$$ g(s)= \int_{0}^{\infty} \frac{f(t)}{t+s} $$

then let be $ m_{n} = \int_{0}^{\infty}t^{n}f(t) $

then i get the distributional solution

$$ f(t) = \sum_{n=0}^{\infty}\frac{m_{n}}{n!}(-1)^{n}\delta^{n} (t) $$ (1)

so , if i insert this solution i get the asymptotics

$$ g(s)= \sum_{n=0}^{\infty} (-1)^{n}\frac{m_{n}}{s^{n+1}} $$

so does the solution (1) have any utility in the moment problem or something ? thanks