Limit of $\frac {n^n}{n!}$ [duplicate]

I have to prove that

$$\lim_{n\to \infty} \frac {n^n} {n!}=\infty$$

I've tried to look for a lower bound that also converges to $\infty$ (I don't know if I'm explainig myself correctly), but I haven't found one yet. Applying L'Hôpital is way too complicated in $n!$, and the epsilon proof does not work as I have no way whatsoever of finding N.

Any ideas?

marked as duplicate by MJD, Thomas, Daniel Fischer♦, froggie, egregFeb 14 '14 at 20:23

• ${n^n\over n!}=\color{maroon}{{n\over\vphantom{1} n}{n\over n-1}\cdots {n\over 2}}{n\over 1}\ge \color{maroon}1\cdot n$. – David Mitra Feb 14 '14 at 18:48
• intuitively, $$\frac{n^n}{n!}=\frac{n\cdot{n}\cdot{...}\cdot{n}\cdot n}{n(n-1)...2\cdot{1}}$$ – Eleven-Eleven Feb 14 '14 at 18:49
• Alternatively you can prove that $\frac{n!}{n^n} \to 0$, hence the reciprocal tends to $\infty$ – Alex Feb 14 '14 at 18:51
• @Alex that´s the limit I wanted in the first place, but for some reason I thought this was easier. – Lessa121 Feb 14 '14 at 18:52
• @DavidMitra Thanks! – Lessa121 Feb 14 '14 at 18:53

Setting $$a_n=\frac{n^n}{n!},$$ we have $$\frac{a_{n+1}}{a_n}=\frac{(n+1)^{n+1}}{(n+1)!}\cdot\frac{n!}{n^n}=\frac{(n+1)^n\cdot n!}{n!\cdot n^n}=\frac{(n+1)^n}{n^n}=\left(1+\frac{1}{n}\right)^n \quad \forall n.$$ Since $$\lim_n\frac{a_{n+1}}{a_n}=\lim_n\left(1+\frac{1}{n}\right)^n=e>2,$$ there is an $N \in \mathbb{N}$ such that $$\frac{a_{n+1}}{a_n}>2 \quad \forall n\ge N.$$ It follows that $$a_n\ge 2^{n-N}a_N \quad \forall n\ge N,$$ thus $$\lim_na_n\ge \lim_n2^{n-N}a_N,$$ i.e. $\lim_na_n=\infty$.