Solve Inequality Homework: A Hint Needed

[cupcakes]

Homework Statement

To prove the inequality (attached)

Homework Equations

The Attempt at a Solution

I tried factoring out a 2 from each of the even terms in the denominator. This allowed me to cancel out all the terms (odd) on the numerator up to 1005.

Leaves me with:

$$\frac{1}{2^{1006}}\frac{1007*1009*...*2009*2011}{2*4*...*1004*1006} < \frac{1}{\sqrt{2010}}$$

I don't know how to continue after this point. Can someone please give me a hint? Thanks.

 

[Simon Bridge]

um, that would be: $$\prod_{n=1}^{1006} \frac{2n-1}{2n} < \frac{1}{\sqrt{2010}}$$ can also be written in terms of double factorials: $$\frac{(2k-1)!}{(2k)!}: k=1006$$

Then you have things like: $$(2k-1)!=\frac{(2k)!}{2^k k!} \qquad \text{and} \qquad (2k)!=2^k k!$$

http://en.wikipedia.org/wiki/Factorial#Double_factorial
... the identities will be standard proofs you can look up.

notice that 2010 = 2(k-1)