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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 $1/\left(1+\frac{1}{n}\right)^n$

  • 3
    Stirling's approximation http://en.wikipedia.org/wiki/Stirling's_approximation2012-09-24
  • 4
    @Marvis I suspect that Stirling's approximation would not be in the spirit of the question.2012-09-24
  • 1
    Related: http://math.stackexchange.com/questions/28476/finding-the-limit-of-frac-n-sqrtnn2012-12-14
  • 0
    http://math.stackexchange.com/questions/28476/finding-the-limit-of-frac-n-sqrtnn2014-02-21

5 Answers 5

1

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}{

\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*}

51

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 $$\int_0^1 \log x\,dx.$$

  • 2
    This is really neat!2012-09-25
29

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

  • 0
    Yes!! Thank you!!2012-09-25
  • 1
    I cannot follow what you are doing in your 3rd equal sign.2012-09-25
  • 0
    @MartinArgerami: Notice that $n!=1\cdot 2\cdots\cdot n$.2012-09-25
  • 1
    Yes, now I see it. I though it was a limit manipulation and not just algebra. Thanks.2012-09-25
21

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

  • 0
    For the inequality used in this answer see [this post](http://math.stackexchange.com/questions/69386/inequality-involving-limsup-and-liminf). (And also other posts listed there among linked questions.)2014-06-24
6

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

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