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Solve inequality

  1. Sep 12, 2012 #1
    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:

    [tex]
    \frac{1}{2^{1006}}\frac{1007*1009*...*2009*2011}{2*4*...*1004*1006} < \frac{1}{\sqrt{2010}}
    [/tex]

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

    Attached Files:

  2. jcsd
  3. Sep 12, 2012 #2

    Simon Bridge

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

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

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

    notice that 2010 = 2(k-1)
     
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