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Question Number 204174 by patrice last updated on 08/Feb/24

Answered by Faetmaaa last updated on 27/Feb/24

$$3)-$$$$\mathrm{\forall }n\in {\mathbf{N}}^{★}$$$$S_n=\frac{1}{n}\underset{k=1}{\sum ^n}\ln \left(\frac{k}{n}\right)$$$$=\frac{1-0}{n}\underset{k=1}{\sum ^n}\ln (0+k\frac{1-0}{n})$$$$\underset{n\to \mathrm{\infty }}{\to }{\int }_0^1\ln \left(x\right)\mathrm{d}x$$$$={\left[x(\ln x-1)\right]}_0^1$$$$=-1$$

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