Calculation of $\lim_{n\rightarrow \infty}\frac{n!\cdot e^{n}}{n^n\sqrt{n}}$

Asked Jun 04 '19 at 07:05

Active Jun 04 '19 at 07:22

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Calculation of limit $\displaystyle \lim_{n\rightarrow \infty}\frac{n!\cdot e^{n}}{n^n\sqrt{n}}$

Plan Stirling approximation $$n!\approx \bigg(\frac{n}{e}\bigg)^n\sqrt{2\pi n}$$ for large $n$

$$\lim_{n\rightarrow \infty}\frac{n^n}{e^n}\cdot \sqrt{2\pi n}\cdot \frac{1}{\sqrt{n}}=\sqrt{2\pi}$$

How would I evaluate the limit without Stirling's approximation?

edited Jun 04 '19 at 07:22

user3184807

asked Jun 04 '19 at 07:05

jacky

2

I don't see any other way to get a handle on $n!$, why do you want to do that? – plus1 Jun 04 '19 at 07:16
5

I would go one step further: this limit being $\sqrt{2\pi}$ is immediately equivalent to Stirling's approximation. Maybe you should be looking for proofs of Stirling's approximation? – Theo Bendit Jun 04 '19 at 07:18

0 Answers0