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Math Help - Just for Fun

  1. #1
    MHF Contributor chiph588@'s Avatar
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    Just for Fun

    This problem is just for fun, meant as a challenge.

    Calculate  \lim_{n \to \infty} \frac{2^{2n-1}((n-1)!)^2 \sqrt{n}}{(2n-1)!} .
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  2. #2
    MHF Contributor Bruno J.'s Avatar
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    The answer is \sqrt \pi!

    The solution is easy : Gamma duplication formula!
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  3. #3
    MHF Contributor chiph588@'s Avatar
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    Good job!
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  4. #4
    MHF Contributor Bruno J.'s Avatar
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    Thank you for the problem!
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  5. #5
    MHF Contributor chiph588@'s Avatar
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    How did you come about your solution? I thought it would be harder because a lot cancels out so it doesn't resemble the duplication formula.
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  6. #6
    MHF Contributor Bruno J.'s Avatar
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    I don't know, I just thought of it right away. I think 2^{2n-1} gave it away!
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  7. #7
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    i'm sure you guys know that applying Strling's formula will make the problem very straightforward.
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