Explain what is meant by an asymptotic power series about $x=a$ for a real function $f(x)$ of a real variable. Show that a convergent power series is also asymptotic.

Show further that an asymptotic power series is unique (assuming that it exists).

Let the function $f(t)$ be defined for $t \geqslant 0$ by

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

By suitably expanding the denominator of the integrand, or otherwise, show that, as $t \rightarrow 0_{+}$,

$$f(t) \sim \sum_{k=0}^{\infty}(-1)^{k} 1.3 \ldots(2 k-1) t^{k}$$

and that the error, when the series is stopped after $n$ terms, does not exceed the absolute value of the $(n+1)$ th term of the series.