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Math Help - Limiting behavior of a sequence

  1. #1
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    Limiting behavior of a sequence

    The question:

    Describe the limiting behaviour of the following sequence. If the sequence converges, then state its limit.

    \frac{(2n)!}{(n!)^2}

    I'm not sure how to evaluate this. I tried get some intuition of what's going on, but I'm struggling. Any advice?
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  2. #2
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by Glitch View Post
    The question:

    Describe the limiting behaviour of the following sequence. If the sequence converges, then state its limit.

    \frac{(2n)!}{(n!)^2}

    I'm not sure how to evaluate this. I tried get some intuition of what's going on, but I'm struggling. Any advice?
    Hint: Consider using Stirling's approximation for n!.

    Edit: Stirling's formula could be overkill for this. You may want to consider applying the ratio test for sequences instead.

    So let a_n=\dfrac{(2n)!}{(n!)^2}. Now compute \lim\limits_{n\to\infty}\left|\dfrac{a_{n+1}}{a_n}  \right|\rightarrow L. If L<1, then its convergent.

    Can you proceed?
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  3. #3
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    Hmm, the ratio test seems to be much further into this chapter, which is beyond this problem set. I'll try it anyway, however It'd be interesting to know if there's another method.
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  4. #4
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by Glitch View Post
    Hmm, the ratio test seems to be much further into this chapter, which is beyond this problem set. I'll try it anyway, however It'd be interesting to know if there's another method.
    If that's the case then, I would probably go along with my first suggestion and use Stirling's approximation: n!\sim\sqrt{2\pi n}e^{-n}n^n.

    So \lim\limits_{n\to\infty}\dfrac{(2n)!}{(n!)^2}\sim\  lim\limits_{n\to\infty}\dfrac{\sqrt{4\pi n}e^{-2n}(2n)^{2n}}{\left(\sqrt{2\pi n}e^{-n}n^n\right)^2}=\ldots

    Can you proceed?
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  5. #5
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    Ahh yep, that was fairly easy to work out after doing a bit of algebra. Thanks.

    Odd that Stirling's approx. isn't mentioned in this text either. I think they wanted us to solve it via intuition. 0_o
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  6. #6
    MHF Contributor FernandoRevilla's Avatar
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    An alternative:

    \dfrac{(2n)!}{(n!)^2}=\dfrac{ (n+1)\cdot(n+2)\cdot \ldots\cdot (2n)}{1\cdot 2\cdot \ldots\cdot n}=\displaystyle\prod_{k=1}^{n}\left(1+\frac{n}{k}  \right)\geq 2^n

    and

    \displaystyle\lim_{n \to{+}\infty}{2^n}=+\infty

    so, the limit of the given sequence is +\infty


    Fernando Revilla
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  7. #7
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    Quote Originally Posted by Chris L T521 View Post
    Hint: Consider using Stirling's approximation for n!.

    Edit: Stirling's formula could be overkill for this. You may want to consider applying the ratio test for sequences instead.

    So let a_n=\dfrac{(2n)!}{(n!)^2}. Now compute \lim\limits_{n\to\infty}\left|\dfrac{a_{n+1}}{a_n}  \right|\rightarrow L. If L<1, then its convergent.

    Can you proceed?
    The ratio test is test for convergencd of series, not sequences.

    If would write \frac{(2n)!}{(n!)^2}= \frac{n!(n+1)(n+2)\cdot\cdot\cdot (2n)}{n! n!}= \frac{n+1}{1}\frac{n+2}{2}\frac{n+3}{3}\cdot\cdot\  cdot\frac{2n}{n}
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  8. #8
    MHF Contributor FernandoRevilla's Avatar
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    I think Chris L T521 meant to try the ratio test to see if the limit is L<1 . In that case we would deduce that the limit of the sequence is 0 :

    \displaystyle\lim_{n \to{+}\infty}{\left |{\dfrac{a_{n+1}}{a_n}}\right |}=L<1 \Rightarrow \displaystyle\sum_{n=1}^{+\infty}a_n\;\textrm{conv  ergent}\;\Rightarrow

     l=\displaystyle\lim_{n \to{+}\infty}{a_n}=0\Rightarrow (a_n)\;\textrm{convergent}


    Fernando Revilla
    Last edited by FernandoRevilla; January 23rd 2011 at 11:48 AM.
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  9. #9
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by FernandoRevilla View Post
    I think Chris L T521 meant to try the ratio test to see if the limit is L<1 . In that case we would deduce that the limit of the sequence is 0 :


    \displaystyle\lim_{n \to{+}\infty}{\left |{\dfrac{a_{n+1}}{a_n}}\right |}=L<1 \Rightarrow \displaystyle\sum_{n=1}^{+\infty}a_n\;\textrm{conv  ergent}\;\Rightarrow l=\displaystyle\lim_{n \to{+}\infty}{a_n}=0\Rightarrow (a_n)\;\textrm{convergent}


    Fernando Revilla
    Yes, that's what I was trying to get at. Sorry that I wasn't explicit about that earlier. I was in essence using the following theorem:

    Theorem: Suppose that (s_n) is a sequence of positive terms and that the sequence of ratios (s_{n+1}/s_n) converges to L. If L<1, then \lim s_n = 0.
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