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Question Number 140513 by mathocean1 last updated on 08/May/21
n ∈ N^∗  and k ∈ N^∗ .  Given 0≤k≤n−1.  Show that   (1/n)ln(1+(k/n))≤∫_(1+(k/n)) ^(1+((k+1)/n)) lnx dn≤(1/n)ln(1+((k+1)/n))
$$\mathrm{n}\:\in\:\mathbb{N}^{\ast} \:\mathrm{and}\:\mathrm{k}\:\in\:\mathbb{N}^{\ast} . \\ $$$$\mathrm{Given}\:\mathrm{0}\leqslant\mathrm{k}\leqslant\mathrm{n}−\mathrm{1}. \\ $$$$\mathrm{Show}\:\mathrm{that}\: \\ $$$$\frac{\mathrm{1}}{\mathrm{n}}\mathrm{ln}\left(\mathrm{1}+\frac{\mathrm{k}}{\mathrm{n}}\right)\leqslant\underset{\mathrm{1}+\frac{\mathrm{k}}{\mathrm{n}}} {\overset{\mathrm{1}+\frac{\mathrm{k}+\mathrm{1}}{\mathrm{n}}} {\int}}\mathrm{lnx}\:\mathrm{dn}\leqslant\frac{\mathrm{1}}{\mathrm{n}}\mathrm{ln}\left(\mathrm{1}+\frac{\mathrm{k}+\mathrm{1}}{\mathrm{n}}\right) \\ $$

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