Stirling's Approximation but Weaker

Fix \(k\in \mathbb N\). According to the power series representation of the exponential function we should have:

\[\begin{align*} e^n &= 1 + n + \frac{n^2}{2!} + \cdots + \frac{n^n}{n!} + \frac{n^{n+1}}{(n+1)!} + \cdots\\ &\gt \frac{n^n}{n!}\left(1 + \frac {n}{n+1} + \frac {n^2}{(n+1)(n+2)}+ \cdots\right)\\ &\gt \frac{n^n}{n!}\left(1 + \frac {n}{n+1} + \cdots + \frac {n^k}{(n+1)\cdots(n+k)}\right).\tag{1}\label{eq:1} \end{align*}\]

Therefore, \ref{eq:1} gives

\[\liminf_{n\to \infty} \frac{e^nn!}{n^n} \geq k+1 > k\]

and the conclusion follows from the fact that \(k\) was an arbitrary number.