# Solve inequality

1. Sep 12, 2012

### cupcakes

1. The problem statement, all variables and given/known data
To prove the inequality (attached)

2. Relevant equations
3. 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.

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2. Sep 12, 2012

### 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)