Find $\lim_{n\to \infty}((n+1)!)^{\frac{1}{n+1}}-((n)!)^{\frac{1}{n}}.$

Question

Find $\lim_{n\to \infty}((n+1)!)^{\frac{1}{n+1}}-((n)!)^{\frac{1}{n}}.$

We need to deal the limit $\lim_{n\to \infty} \frac{\log(1)+\log(2)+...+\log(n)}{n}$. We know that $\lim_{n\to \infty} \log(n)=\infty \implies \lim_{n\to \infty} \frac{\log(1)+\log(2)+...+\log(n)}{n}=\infty$(since, By Cauchy's first theorem on limit). Hence we get $\infty-\infty$. How do I show that there exists finite limit?

Let me check. But the limit is infinity. – Dec 06 '18 at 16:01 — , Dec 06 '18 at 16:01
The limit should be $1/e$. – Paramanand Singh Dec 06 '18 at 16:04 — Paramanand Singh, Dec 06 '18 at 16:04
How do we get?Can you give some hint? – Dec 06 '18 at 16:05 — , Dec 06 '18 at 16:05

Mark Viola · Accepted Answer · 2018-12-06T20:46:58.983

1

HINT:

Using Stirling's Formula we have

$$\begin{align} \left((n+1)!\right)^{1/(n+1)}-\left(n!\right)^{1/n}&=\left(\left(\sqrt{2\pi(n+1)}\left(\frac{n+1}{e}\right)^{n+1}\right)\left(1+O(1/n)\right)\right)^{1/(n+1)}\\\\ &-\left(\left(\sqrt{2\pi n}\left(\frac{n}{e}\right)^{n}\right)\left(1+O(1/n)\right)\right)^{1/n} \end{align}$$

Can you finish now?

Alternatively, I provided a different, less "brute force" approach in THIS ANSWER.

edited Dec 06 '18 at 20:46

answered Dec 06 '18 at 16:05

Mark Viola

179,405

This it is not convincing at all. Did you really check that this gets somewhere ? – Ewan Delanoy Dec 06 '18 at 16:13
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@EwanDelanoy First of all, it is a HINT only. And yes, I did check this and found that the limit is simply $1/e$. – Mark Viola Dec 06 '18 at 16:15
@EwanDelanoy Alternatively, I provided a different, less "brute force" approach in THIS ANSWER. – Mark Viola Dec 06 '18 at 16:22
Is your sirling formula correct? $\sqrt{2 \pi n}$ in the numerator? – Dec 06 '18 at 17:23
your alternate approach is very excellent :) – Dec 06 '18 at 17:25
@Unknownx I've corrected the typographical error. Thank you for alerting me. – Mark Viola Dec 06 '18 at 20:47
And pleased you liked the other approach I posted quite a while ago. ;-) – Mark Viola Dec 06 '18 at 20:48

Find $\lim_{n\to \infty}((n+1)!)^{\frac{1}{n+1}}-((n)!)^{\frac{1}{n}}.$

1 Answers1