Write the summand as $a_n=x^n/[\Gamma(n)n^n]$. By <Stirling formula>,
$$
a_n\sim\frac{n^{1/2}}{\sqrt{2\pi}}
\exp\left[n(\log x+1-2\log n)\right].
$$
Set
$$
N=\sqrt{\frac xe},
\qquad n=Nt.
$$
Then the exponential phase is
$$
n(\log x+1-2\log n)
=Nf(t),
\qquad
f(t)=2t(1-\log t).
$$
It has a unique maximum at $t=1$, where $f(1)=2$ and $f''(1)=-2$. 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 gives
$$
S\sim
N\frac{\sqrt N}{\sqrt{2\pi}}e^{2N}
\sqrt{\frac{2\pi}{2N}}
=\frac{N}{\sqrt2}e^{2N}.
$$
Hence
$$
\boxed{
S\sim\sqrt{\frac{x}{2e}}
\exp\left(2\sqrt{\frac xe}\right)}
\qquad(x\to\infty).
$$

Solution

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

{c}

The discrete Laplace method estimates a sum whose positive terms concentrate near the maximum of a smooth exponent. When the Gaussian concentration width contains asymptotically many lattice points, the sum is a Riemann-sum version of <Laplace's method> with the corresponding lattice-spacing factor.

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