# Showing that $\frac{\sqrt[n]{n!}}{n}$ $\rightarrow \frac{1}{e}$ [duplicate]

Show:$$\lim_{n\to\infty}\frac{\sqrt[n]{n!}}{n}= \frac{1}{e}$$

So I can expand the numerator by geometric mean. Letting $C_{n}=\left(\ln(a_{1})+...+\ln(a_{n})\right)/n$. Let the numerator be called $a_{n}$ and the denominator be $b_{n}$ Is there a way to use this statement so that I could force the original sequence into the form of $1/\left(1+\frac{1}{n}\right)^n$

Have not found a way to rewrite your expression to get the desired result. However, here is a suggested approach.

Maybe rewrite the left-hand side as $$\sqrt[n]{\frac{n!}{n^n}}.$$

Take the logarithm. We get $$\frac{1}{n}\left(\log\left(\frac{1}{n}\right)+ \log\left(\frac{2}{n}\right)+\log\left(\frac{3}{n}\right)+\cdots+\log\left(\frac{n}{n}\right)\right).$$

Now think of the above sum as a Riemann sum for the not quite proper integral $$\int_0^1 \log x\,dx.$$

• This is really neat! Sep 25, 2012 at 0:01
• This is very impressive. How did you come with it? Like what's the thought process behind the solution Jun 14, 2019 at 8:00
• What does it mean by "not quite proper"? Isn't it improper? Dec 11, 2019 at 9:32

I would like to use the following lemma:

If $\lim_{n\to\infty}a_n=a$ and $a_n>0$ for all $n$, then we have $$\lim_{n\to\infty}\sqrt[n]{a_1a_2\cdots a_n}=a \tag{1}$$

Let $a_n=(1+\frac{1}{n})^n$, then $a_n>0$ for all $n$ and $\lim_{n\to\infty}a_n=e$. Applying ($*$) we have \begin{align} e&=\lim_{n\to\infty}\sqrt[n]{a_1a_2\cdots a_n}\\ &=\lim_{n\to\infty}\sqrt[n]{\left(\frac{2}{1}\right)^1\left(\frac{3}{2}\right)^2\cdots\left(\frac{n+1}{n}\right)^n}\\ &=\lim_{n\to\infty}\sqrt[n]{\frac{(n+1)^n}{n!}}\\&= \lim_{n\to\infty}\frac{n+1}{\sqrt[n]{n!}}=\lim_{n\to\infty}\frac{n}{\sqrt[n]{n!}}+\lim_{n\to\infty}\frac{1}{\sqrt[n]{n!}}=\lim_{n\to\infty}\frac{n}{\sqrt[n]{n!}} \end{align}\tag{2} where we use (1) in the last equality to show that $\lim_{n\to\infty}\frac{1}{\sqrt[n]{n!}}=0.$

It follows from (2) that $$\lim_{n\to\infty}\frac{\sqrt[n]{n!}}{n}=\frac{1}{e}.$$

• Yes!! Thank you!! Sep 25, 2012 at 0:39
• I cannot follow what you are doing in your 3rd equal sign. Sep 25, 2012 at 4:18
• @MartinArgerami: Notice that $n!=1\cdot 2\cdots\cdot n$.
– user9464
Sep 25, 2012 at 15:01
• Yes, now I see it. I though it was a limit manipulation and not just algebra. Thanks. Sep 25, 2012 at 15:03

If $a_n \geq 0$, then the following inequality holds:

$$\liminf_{n\to\infty} \frac{a_{n+1}}{a_n} \leq \liminf_{n\to\infty} \sqrt[n]{a_n} \leq \limsup_{n\to\infty} \sqrt[n]{a_n} \leq \limsup_{n\to\infty} \frac{a_{n+1}}{a_n}.$$

Now let $a_n = n! / n^n$. Then it follows that

$$\frac{a_{n+1}}{a_n} = \frac{\frac{(n+1)!}{(n+1)^{n+1}}}{\frac{n!}{n^n}} = \frac{1}{\left(1+\frac{1}{n}\right)^n},$$

and hence

$$\liminf_{n\to\infty} \frac{a_{n+1}}{a_n} = \limsup_{n\to\infty} \frac{a_{n+1}}{a_n} = \frac{1}{e}.$$

This proves that $\sqrt[n]{a_n} \to e^{-1}$ .

• For the inequality used in this answer see this post. (And also other posts listed there among linked questions.) Jun 24, 2014 at 7:56

It's straightforward if you use Cesaro-Stolz theorem and then the celebre Lalescu's limit.

$$\lim_{n\to\infty}\frac{\sqrt[n]{n!}}{n}= \lim_{n\to\infty} \sqrt[n+1]{(n+1)!}-\sqrt[n]{(n)!}=\frac{1}{e}.$$

Here is a rather direct calculation of the limit using squeezing. It needs

• $\ln k < \int_k^{k+1}\ln x \; dx < \ln (k+1)$
• $\int \ln x \; dx = x(\ln x - 1) \color{grey}{+C}$
• $\lim_{n \rightarrow \infty}\frac{\ln (n+1)}{n} = 0$

Set $$x_n = \ln \frac{\sqrt[n]{n!}}{n} = \frac{1}{n}\sum_{k=1}^n \ln k - \ln n$$ So, to show is $\color{blue}{x_n \stackrel{n\rightarrow \infty}{\longrightarrow} -1}$. We have $$\int_1^n \ln x \; dx < \sum_{k=1}^{n-1} \ln (k+1) =\sum_{k=1}^n \ln k < \int_1^{n+1}\ln x \; dx$$ We now squeeze: $$\color{blue}{L_n} := \frac{1}{n}\int_1^n \ln x \; dx - \ln n \color{blue}{<x_n < } \frac{1}{n} \int_1^{n+1}\ln x \; dx - \ln n =: \color{blue}{R_n}$$

\begin{align*} \color{blue}{L_n} & = \frac{1}{n}\left( n(\ln n - 1) +1 \right) - \ln n \\ & = -1 +\frac{1}{n} \color{blue}{\stackrel{n \rightarrow \infty}{\longrightarrow} -1} \\ & \\ \color{blue}{R_n} & = \frac{1}{n}\left( (n+1)(\ln (n+1) - 1) +1 \right) - \ln n \\ & = \left( 1 + \frac{1}{n} \right) \ln (n+1) - \left( 1 + \frac{1}{n} \right) + \frac{1}{n} - \ln n \\ & = -1 + \ln \left( 1+\frac{1}{n} \right) + \frac{\ln (n+1)}{n} \color{blue}{\stackrel{n \rightarrow \infty}{\longrightarrow} -1} \end{align*}