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Math Help - Limit of a Sequence

  1. #1
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    Limit of a Sequence

    I'm having trouble with the sequence: a_n=\frac{1*3*5*...*(2n-1)}{n!}

    It seems that the limit must exists since each factor a_n=\frac{1}{1}*\frac{3}{2}*\frac{5}{3}*\frac{2n-1}{n} has a finite limit.
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  2. #2
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    Write 1*3*5*...(2n-1) as \frac{1*2*3*4*5*...*(2n-1)(2n)}{2*4*6*...**(2n)}= \frac{(2n)!}{((1)(2))(2(2))*(3(2))*...*(n(2)) =\frac{(2n)!}{2^n n!}.

    Now, a_n= \frac{(2n)!}{2^n(n!)^2}.
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  3. #3
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    Thanks, but you have a latex error.
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  4. #4
    MHF Contributor chisigma's Avatar
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    It is easy to see that is...

    a_{n+1} = a_{n} \frac{2n+1}{n+1} (1)

    ... so that is...

    \lim_{n \rightarrow \infty} \frac{a_{n+1}}{a_{n}} = 2 (2)

    ... and that means that is...

    \lim_{n \rightarrow \infty} a_{n} = \infty (3)

    Kind regards

    \chi \sigma
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