# Inequality with factorial

1. Apr 11, 2012

### cupcakes

1. The problem statement, all variables and given/known data
$\frac{1^2*3^2*5^2...(2n-1)^2}{2^2*4^2*6^2...(2n)^2}<\frac{1}{2n+1}$

Edit: Must be proven without using induction.

2. Relevant equations
3. The attempt at a solution
I understand the LHS is the same thing as

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

And (2n)!! = $k!2^k$ & (2n-1)!! = $\frac{(2k)!}{k!2^k}$

I've tried substituting and it doesn't seem to help. Any ideas? Thanks.

Last edited: Apr 11, 2012
2. Apr 11, 2012

### scurty

I was able to prove this by induction, try it out!

3. Apr 11, 2012

### SammyS

Staff Emeritus
There are some typos there.

$\displaystyle (2k)!!=k!2^k$

$\displaystyle (2k-1)!!=\frac{(2k-1)!!\ (2k)!!}{(2k)!!}=\frac{(2k)!}{k!2^k}$

4. Apr 11, 2012

### cupcakes

I forgot to mention that the problem states that it must be proven without using induction. :(

Thanks Sammy
Does anyone have any other idea or hint that does not involve induction?

5. Apr 12, 2012

### cupcakes

I've solved it.