limt with factorial

**diroga** · June 9th 2009, 10:34 AM

find the limit of $\frac{(2n - 1)!}{(2n + 1)!}$ as n approaches infinity.

L'hop wont work b.c you cant take the derivative of a factorial. I wrote out the first three terms and they appear to be decreasing and approaching zero.

**flyingsquirrel** · June 9th 2009, 10:53 AM

Hi,

We have $(2n+1)!=(2n+1)\times(2n)\times (2n-1)!$ so

$\frac{(2n-1)!}{(2n+1)!}=\frac{(2n-1)!}{(2n+1)\times(2n)\times (2n-1)!}=\frac{1}{(2n+1)\times(2n)}$

Now you can compute the limit, can't you?

**Amer** · June 9th 2009, 11:01 AM

Originally Posted by diroga

find the limit of $\frac{(2n - 1)!}{(2n + 1)!}$ as n approaches infinity.

L'hop wont work b.c you cant take the derivative of a factorial. I wrote out the first three terms and they appear to be decreasing and approaching zero.

$lim_{n\rightarrow\infty} \frac{(2n-1)!}{(2n+1)!}=lim_{n\rightarrow\infty} \frac{(2n-1)!}{(2n+1)(2n)(2n-1)!}$ $=\frac{{\color{red}(2n-1)!}}{(2n+1)(2n){\color{red}(2n-1)!}}=lim_{n\rightarrow\infty}\frac{1}{(2n+1)(2n)} =0$

**diroga** · June 9th 2009, 02:06 PM

I dont see how you could do this
$<br /> (2n+1)!=(2n+1)\times(2n)\times (2n-1)!<br />$

I guess when it comes to limits you got be good with algebra and property tricks

**Ruun** · June 9th 2009, 02:14 PM

$0!=1$

$1!=1$

$2!=2\cdot 1$

$3!=3\cdot 2 \cdot 1= 3 \cdot 2!$

$n!=n \cdot (n-1) \cdot (n-2) \cdot ... \cdot 3 \cdot 2 \cdot 1$

$(2n+1)!=(2n+1) \cdot (2n) \cdot (2n - 1) \cdot ... \cdot 1 = (2n+1) \cdot (2n) \cdot (2n-1)!$

**HallsofIvy** · June 9th 2009, 02:23 PM

Originally Posted by diroga

I dont see how you could do this
$<br /> (2n+1)!=(2n+1)\times(2n)\times (2n-1)!<br />$

I guess when it comes to limits you got be good with algebra and property tricks

Or learn the definitions?

**diroga** · June 9th 2009, 08:39 PM

Originally Posted by Ruun

$0!=1$

$1!=1$

$2!=2\cdot 1$

$3!=3\cdot 2 \cdot 1= 3 \cdot 2!$

$n!=n \cdot (n-1) \cdot (n-2) \cdot ... \cdot 3 \cdot 2 \cdot 1$

$(2n+1)!=(2n+1) \cdot (2n) \cdot (2n - 1) \cdot ... \cdot 1 = (2n+1) \cdot (2n) \cdot (2n-1)!$

thanks

Originally Posted by HallsofIvy

Or learn the definitions?

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