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$\lim\limits_{n \to \infty} \frac{1}{n}[(n+1)(n+2) \ldots (n+n)]^{\frac{1}{n}}=(1+\frac{1}{n})^{\frac{1}{n}}(1+\frac{2}{n})^{\frac{2}{n}}\ldots (1+\frac{n}{n})^{\frac{1}{n}} $.

Each of the terms in the product on the RHS approach 1 as $n \to \infty$. I am guessing this limit will have the exponential funtion in some form. For now I'm stuck. Any help!

JimmyK4542
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Santosh kumar Bhal
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  • Another idea, other than the linked answers, is to use Stirling's Approximation, while noting that the this is equivalent to $\frac{1}{n}(\frac{(2n)!}{n!})^{(1/n)}$ – Moya Jan 20 '16 at 05:20

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