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Question Number 5513 by 314159 last updated on 17/May/16

If n>1, prove by mathematical induction that  n((n+1)^(1/n) −1) < 1+(1/2)+(1/3)+(1/4)+...(1/n).

$${If}\:{n}>\mathrm{1},\:{prove}\:{by}\:{mathematical}\:{induction}\:{that} \\ $$ $${n}\left(\left({n}+\mathrm{1}\right)^{\frac{\mathrm{1}}{{n}}} −\mathrm{1}\right)\:<\:\mathrm{1}+\frac{\mathrm{1}}{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{3}}+\frac{\mathrm{1}}{\mathrm{4}}+...\frac{\mathrm{1}}{{n}}. \\ $$

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