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Product Proof

  1. Apr 2, 2008 #1

    gop

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    1. The problem statement, all variables and given/known data

    Proof that for n>2 and n is a natural number it holds that

    [tex]\prod_{k=1}^{n}\frac{k^{2}+2}{k^{2}+1}<3[/tex]

    and
    [tex]\prod_{k=1}^{n}\frac{k^{2}+2}{k^{2}+1}<\frac{3n}{n+1}[/tex]

    2. Relevant equations



    3. The attempt at a solution

    My best approach was to split the product over the fraction and then to arrive at a statement that looks like

    [tex]\prod_{k=2}^{n}k^{2}+2<\prod_{k=1}^{n}k^{2}+1[/tex]

    I then tried to prove by induction that this statement holds but that doesn't really work. The best result I got (for n+1) is then

    [tex](\prod_{k=2}^{n}k^{2}+2)<(\prod_{k=1}^{n}k^{2}+1)\cdot\frac{n^{2}+2n+2}{n^{2}+2n+3}[/tex]

    But I can't do anything usefuel with that...
     
  2. jcsd
  3. Apr 2, 2008 #2

    Avodyne

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    Science Advisor

    You could try writing the product as the exponential of a sum, and then bounding the sum by an integral.
     
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