Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 2 / 30A (Asymptotic Methods)

The statementas $x\to x_0$ means that for every $N\ge 0$,

Write the remainders after order $N$ as $r_N=o((x-x_0{)}^N)$ for $f$ and $s_N=o((x-x_0{)}^N)$ for $g$. ThenHence $h=f+g$ has the termwise sum of the two asymptotic expansions.

For every $N$,Integrating the finite sum givesFor every $\epsilon >0$, sufficiently close to $x_0$ one has $|r_N(\xi )|\le \epsilon |\xi -x_0{|}^N$, so the last integral is $o(|x-x_0{|}^{N+1})$. This proves the integrated expansion.

LetSince $d(e^{-t^2})=-2t{e}^{-t^2}dt$, repeated integration by parts givesThe final remainder estimate follows by one further integration by parts, or by bounding the remaining positive integral by its leading boundary term. Therefore the requested first three terms are