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Math Help - Use Induction to prove

  1. #1
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    Use Induction to prove

    By Induction prove
    \Gamma (n+\frac{1}{2})\ =\ \frac{1.3.5\cdots (2n-1)}{2^n}\sqrt{\pi}
    Recall \Gamma (\frac{1}{2})\ =\ \sqrt{\pi}

    I know proof by induction is something to do with putting a +1 by the n's however i am fairly clueless as to this example. Thanks.

    ps: the dots between 1.3.5 represent multiplying 1 3 and 5. and the dots after are meant to symbolise a series continuing. i'm also a bit confused as to what that series is... 1, 3, 5 perhaps it's just a difference of two each time?
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Re: Use Induction to prove

    Using a well known property of the gamma function, write \Gamma\left(n+1+\frac{1}{2}\right)=\left(n+\frac{1  }{2}\right)\Gamma\left(n+\frac{1}{2}\right) .
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  3. #3
    MHF Contributor chisigma's Avatar
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    Re: Use Induction to prove

    Quote Originally Posted by maxgunn555 View Post
    By Induction prove
    \Gamma (n+\frac{1}{2})\ =\ \frac{1.3.5\cdots (2n-1)}{2^n}\sqrt{\pi}
    Recall \Gamma (\frac{1}{2})\ =\ \sqrt{\pi}

    I know proof by induction is something to do with putting a +1 by the n's however i am fairly clueless as to this example. Thanks.

    ps: the dots between 1.3.5 represent multiplying 1 3 and 5. and the dots after are meant to symbolise a series continuing. i'm also a bit confused as to what that series is... 1, 3, 5 perhaps it's just a difference of two each time?
    Using the basic property of the Gamma Function...

    \Gamma(x+1)=x\ \Gamma(x) (1)

    ... and setting a_{n}= \Gamma(n+\frac{1}{2}) You arrive to write the difference equation...

    a_{n+1}= (n+\frac{1}{2})\ a_{n}\ ,\ a_{0}=\sqrt{\pi} (2)

    As explained in...

    http://www.mathhelpforum.com/math-he...-i-188482.html

    ... the solution of (2) is...

    a_{n}= a_{0}\ \prod_{k=0}^{n-1} \frac{2k+1}{2}= \sqrt{\pi}\ \frac {1 \cdot 3 \cdot ... (2n-1)}{2^{n}} (3)

    Kind regards

    \chi \sigma
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