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Homework Help: Proof that ratio of the products of odd and even numbers converges

  1. Sep 23, 2012 #1
    1. The problem statement, all variables and given/known data
    Show that

    [itex]a_{n} = \frac{1 \cdot 3 \cdot 5 \cdot ... \cdot (2n - 1)}{2 \cdot 4 \cdot 6 \cdot ... \cdot 2n}[/itex]

    converges when n [itex]\rightarrow[/itex]∞

    and n is a natural number

    2. Relevant equations
    None that I can think of.


    3. The attempt at a solution

    This was from an exam and I was pretty much stumped.
     
    Last edited: Sep 23, 2012
  2. jcsd
  3. Sep 23, 2012 #2

    jbunniii

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    What can you say about [itex]a_{n+1}[/itex] vs. [itex]a_{n}[/itex]? Which one is larger?
     
  4. Sep 23, 2012 #3
    Thanks for the hint!

    As far as I understand the problem we start by 1/2 and multiply another rational for each increment increase in n. Meaning that [itex]a_{n+1}[/itex] = [itex]a_{n}[/itex] * (2n)/(2n +1). Since n is a natural number that menas the latter factor is less than one, therefore [itex]a_{n}[/itex] must be greater than [itex]a_{n+1}[/itex]

    I'm pretty sure I can work with this, but right now I need to be away from the computer for an hour or so.

    So far I'm thinking maybe proof by induction to show that each n must be smaller than the next and to use the fact that they're natural numbers to put a lower limit of zero on the product. And then squeeze [itex]a_{n}[/itex] to zero.

    Thanks a lot
     
  5. Sep 23, 2012 #4

    Ray Vickson

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    The denominator is [itex] 2 \cdot 4 \cdot 6 \cdots 2n = 2^n n! [/itex] and the numerator is
    [tex] 1 \cdot 3 \cdot 5 \cdots (2n-1) = \frac{(2n-1)!}{2 \cdot 4 \cdots (2n-2)}
    = \frac{(2n-1)!}{2^{n-1} (n-1)!}.[/tex] Thus,
    [tex] a_n = \frac{(2n-1)!}{2^{n-1} (n-1)!} \cdot \frac{1}{2^n n!}.[/tex] Now use Stirling's formula.

    RGV
     
  6. Sep 23, 2012 #5

    jbunniii

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    You don't need to "squeeze [itex]a_n[/itex] to zero." All the problem asks you to do is to show that the sequence converges. This is easier than finding what value it converges to.

    You established that [itex]a_n[/itex] is a monotonically decreasing sequence. What do you know about convergence of monotone sequences?
     
    Last edited: Sep 23, 2012
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