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Math Help - limt with factorial

  1. #1
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    limt with factorial

    find the limit of \frac{(2n - 1)!}{(2n + 1)!} as n approaches infinity.

    L'hop wont work b.c you cant take the derivative of a factorial. I wrote out the first three terms and they appear to be decreasing and approaching zero.
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  2. #2
    Super Member flyingsquirrel's Avatar
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    Hi,

    We have (2n+1)!=(2n+1)\times(2n)\times (2n-1)! so
    \frac{(2n-1)!}{(2n+1)!}=\frac{(2n-1)!}{(2n+1)\times(2n)\times (2n-1)!}=\frac{1}{(2n+1)\times(2n)}

    Now you can compute the limit, can't you?
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  3. #3
    MHF Contributor Amer's Avatar
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    Quote Originally Posted by diroga View Post
    find the limit of \frac{(2n - 1)!}{(2n + 1)!} as n approaches infinity.

    L'hop wont work b.c you cant take the derivative of a factorial. I wrote out the first three terms and they appear to be decreasing and approaching zero.

    lim_{n\rightarrow\infty} \frac{(2n-1)!}{(2n+1)!}=lim_{n\rightarrow\infty} \frac{(2n-1)!}{(2n+1)(2n)(2n-1)!} =\frac{{\color{red}(2n-1)!}}{(2n+1)(2n){\color{red}(2n-1)!}}=lim_{n\rightarrow\infty}\frac{1}{(2n+1)(2n)}  =0
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  4. #4
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    I dont see how you could do this
    <br />
(2n+1)!=(2n+1)\times(2n)\times (2n-1)!<br />

    I guess when it comes to limits you got be good with algebra and property tricks
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  5. #5
    Member Ruun's Avatar
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    0!=1

    1!=1

    2!=2\cdot 1

    3!=3\cdot  2 \cdot 1= 3 \cdot 2!

    n!=n \cdot (n-1) \cdot (n-2) \cdot  ... \cdot 3 \cdot 2 \cdot 1

    (2n+1)!=(2n+1) \cdot (2n) \cdot (2n - 1) \cdot ... \cdot 1 = (2n+1) \cdot (2n) \cdot (2n-1)!
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  6. #6
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    Quote Originally Posted by diroga View Post
    I dont see how you could do this
    <br />
(2n+1)!=(2n+1)\times(2n)\times (2n-1)!<br />

    I guess when it comes to limits you got be good with algebra and property tricks
    Or learn the definitions?
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  7. #7
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    Quote Originally Posted by Ruun View Post
    0!=1

    1!=1

    2!=2\cdot 1

    3!=3\cdot  2 \cdot 1= 3 \cdot 2!

    n!=n \cdot (n-1) \cdot (n-2) \cdot  ... \cdot 3 \cdot 2 \cdot 1

    (2n+1)!=(2n+1) \cdot (2n) \cdot (2n - 1) \cdot ... \cdot 1 = (2n+1) \cdot (2n) \cdot (2n-1)!
    thanks

    Quote Originally Posted by HallsofIvy View Post
    Or learn the definitions?
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