# Weird limit…

1. Aug 2, 2004

### JonF

does...

$$\lim_{n \rightarrow \infty} \frac{n}{(n!)^\frac{1}{n}} = e$$

If not, is it divergent?

2. Aug 2, 2004

### mathwonk

isnt there somethiong called stirlings formula for n! ?? Maybe you could use that and lhopital.

3. Aug 2, 2004

### mathwonk

well i just looked up stirling and it seems to suggest at a quick calculation, not guaranteed, that this limit is e/sqrt(2pi)

4. Aug 2, 2004

### JonF

e/(2pi)^(1/2) aprox= 1.0844

my calc can do the limit up to 200 and it equals about 2.67021... that's why i thought it may = e

5. Aug 3, 2004

### Galileo

With Stirlings approximation: $N!\approx N^Ne^{-N}$, you indeed get:

$$\frac{N}{(N^Ne^{-N})^{\frac{1}{N}}}=\frac{N}{Ne^{-1}}=e$$