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Question Number 140513 by mathocean1 last updated on 08/May/21

$$\mathrm{n}\in {\mathbb{N}}^{\ast }\mathrm{and}\mathrm{k}\in {\mathbb{N}}^{\ast }.$$$$\mathrm{Given}0⩽\mathrm{k}⩽\mathrm{n}-1.$$$$\mathrm{Show}\mathrm{that}$$$$\frac{1}{\mathrm{n}}\ln (1+\frac{\mathrm{k}}{\mathrm{n}})⩽\underset{1+\frac{\mathrm{k}}{\mathrm{n}}}{\stackrel{1+\frac{\mathrm{k}+1}{\mathrm{n}}}{\int }}\mathrm{lnx}\mathrm{dn}⩽\frac{1}{\mathrm{n}}\ln (1+\frac{\mathrm{k}+1}{\mathrm{n}})$$

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