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Math Help - Another limit

  1. #1
    Math Engineering Student
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    Another limit

    Prove that \sqrt[n^2]{\binom{n+1}1\binom{n+2}2\cdots\binom{n+n}n}\to\fr  ac4{\sqrt e} as n\to\infty.
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  2. #2
    Math Engineering Student
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    First off, rewrite that as \exp \left\{ \frac{1}{n^{2}}\prod\limits_{i=1}^{n}{\binom{n+i}i  } \right\} and prove that \frac1{n^2}\prod\limits_{i=1}^{n}{\binom{n+i}i}=\f  rac{1}{n^{2}}\sum\limits_{i=1}^{n}{\sum\limits_{j=  1}^{n}{\ln \frac{i+j}{n}}}-\frac{1}{n}\sum\limits_{i=1}^{n}{\ln \frac{i}{n}}.

    It's a beautiful exercise proving this!
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