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Inequality with factorial

  1. Apr 11, 2012 #1
    1. The problem statement, all variables and given/known data
    [itex]\frac{1^2*3^2*5^2...(2n-1)^2}{2^2*4^2*6^2...(2n)^2}<\frac{1}{2n+1}[/itex]

    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

    [tex]\frac{(2n-1)!!}{(2n)!!}[/tex]

    And (2n)!! = [itex]k!2^k[/itex] & (2n-1)!! = [itex]\frac{(2k)!}{k!2^k}[/itex]

    I've tried substituting and it doesn't seem to help. Any ideas? Thanks.
     
    Last edited: Apr 11, 2012
  2. jcsd
  3. Apr 11, 2012 #2
    I was able to prove this by induction, try it out!
     
  4. Apr 11, 2012 #3

    SammyS

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    There are some typos there.

    [itex]\displaystyle (2k)!!=k!2^k[/itex]

    [itex]\displaystyle (2k-1)!!=\frac{(2k-1)!!\ (2k)!!}{(2k)!!}=\frac{(2k)!}{k!2^k}[/itex]
     
  5. Apr 11, 2012 #4
    I forgot to mention that the problem states that it must be proven without using induction. :(

    Thanks Sammy :smile:
    Does anyone have any other idea or hint that does not involve induction?
     
  6. Apr 12, 2012 #5
    I've solved it. :smile:
     
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