Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 336 / 2

LetThe inequality $|1-e^{-x^2}|\le x^2$ and a split into $0<x<{\epsilon }^2$, ${\epsilon }^2<x<\epsilon$, and $\epsilon <x<1$ show thatThus replacing the exponential by one does not affect any term through $O(1)$.

FactorwherePartial fraction decomposition givesThe required asymptotic expansion follows fromThereforeThe two small roots reveal the same nested scales $x=O({\epsilon }^2)$ and $x=O(\epsilon )$ that a divide-and-conquer asymptotic expansion would match explicitly.

Write the summand as $a_n=x^n/[\Gamma (n)n^n]$. By Stirling formula,SetThen the exponential phase isIt has a unique maximum at $t=1$, where $f(1)=2$ and $f^{\prime \prime }(1)=-2$. The contributing indices satisfy $n-N=O(\sqrt{N})$, so their width tends to infinity and the lattice sum may be replaced by a Riemann sum. The Discrete Laplace method therefore givesHence