(i) Explain what is meant by the assertion: "the series $\sum_{0}^{\infty} b_{n} x^{n}$ is asymptotic to $f(x)$ as $x \rightarrow 0 "$.

Consider the integral

$$I(\lambda)=\int_{0}^{A} e^{-\lambda x} g(x) d x$$

where $A>0, \lambda$ is real and $g$ has the asymptotic expansion

$$g(x) \sim a_{0} x^{\alpha}+a_{1} x^{\alpha+1}+a_{2} x^{\alpha+2}+\ldots$$

as $x \rightarrow+0$, with $\alpha>-1$. State Watson's lemma describing the asymptotic behaviour of $I(\lambda)$ as $\lambda \rightarrow \infty$, and determine an expression for the general term in the asymptotic series.

(ii) Let

$$h(t)=\pi^{-1 / 2} \int_{0}^{\infty} \frac{e^{-x}}{x^{1 / 2}(1+2 x t)} d x$$

for $t \geqslant 0$. Show that

$$h(t) \sim \sum_{k=0}^{\infty}(-1)^{k} 1.3 . \cdots \cdot(2 k-1) t^{k}$$

as $t \rightarrow+0$.

Suggest, for the case that $t$ is smaller than unity, the point at which this asymptotic series should be truncated so as to produce optimal numerical accuracy.