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A Riemann sum II

Evaluate the limit:

    \[\ell =\lim_{n \rightarrow + \infty}\left(\ln\sqrt[n+1]{\frac{n+1}{n}}+\ln\sqrt[n+2]{\frac{n+2}{n}}+\cdots+\ln\sqrt[3n]{\frac{3n}{n}}\right)\]

Solution

We have successively:

    \begin{align*} \ln \prod_{k=1}^{2n} \left(\frac{n+k}{n}\right)^{\frac{1}{n+k}} &= \sum_{k=1}^{2n} \ln \left(\frac{n+k}{n}\right)^{\frac{1}{n+k}} \\ &= \sum_{k=1}^{2n}\frac{1}{n+k} \ln \left(1+\frac{k}{n}\right) \\ &= \frac{1}{n}\sum_{k=1}^{2n}\frac{1}{1+\frac{k}{n}} \ln \left(1+\frac{k}{n}\right) \\ &\rightarrow \int_{0}^{2} \frac{\ln \left ( 1+x \right )}{1+x} \, \mathrm{d}x \\ &= \frac{\ln^2 3}{3} \end{align*}

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