What precisely is meant by the statement that

$$f(x) \sim \sum_{n=0}^{\infty} d_{n} x^{n}$$

as $x \rightarrow 0 ?$

Consider the Stieltjes integral

$$I(x)=\int_{1}^{\infty} \frac{\rho(t)}{1+x t} d t$$

where $\rho(t)$ is bounded and decays rapidly as $t \rightarrow \infty$, and $x>0$. Find an asymptotic series for $I(x)$ of the form $(*)$, as $x \rightarrow 0$, and prove that it has the asymptotic property.

In the case that $\rho(t)=e^{-t}$, show that the coefficients $d_{n}$ satisfy the recurrence relation

$$d_{n}=(-1)^{n} \frac{1}{e}-n d_{n-1} \quad(n \geqslant 1)$$

and that $d_{0}=\frac{1}{e}$. Hence find the first three terms in the asymptotic series.