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A limit with Gamma

Let \Gamma denote the Gamma Euler function. Evaluate the limit

    \[\ell= \lim_{n \rightarrow +\infty} \left[ (n+1) \sqrt[n+1] {\Gamma \left( \frac{1}{n+1} \right)} - n\sqrt[n]{\Gamma \left( \frac{1}{n} \right)} \right]\]

Solution

We might begin by the well known formula

(1)   \begin{equation*}\Gamma \left( \frac{1}{n} \right) = n - \gamma +\mathcal{O} \left( \frac{1}{n} \right) \end{equation*}

Hence

\begin{aligned} \sqrt[n+1]{\Gamma\left(\frac{1}{n+1}\right)}&=\exp\left(\frac{1}{n+1}\ln\Gamma\left(\frac{1}{n+1}\right)\right) \\ &\stackrel{(1)}{=}\exp\left(\frac{1}{n+1}\ln\left(n+1-\gamma+\mathcal O(n^{-1})\right)\right) \\ &=\exp\left(\frac{\ln n}{n}+\mathcal O(n^{-2}\ln n )\right) \\ &= 1+\frac{\ln n}{n}+\mathcal O(n^{-2}\ln^2n) \end{aligned}

We multiply by n+1. We do the same to the other term and then we substract. The result is 1.

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