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Question Number 28537 by abdo imad last updated on 26/Jan/18

prove that   ∀ n∈ N  (1/(√(1+(n+1)^2 ))) ≤ argsh(n+1) −argsh(n)≤ (1/(√(1+n^2 ))) .

$${prove}\:{that}\:\:\:\forall\:{n}\in\:\mathbb{N} \\ $$$$\frac{\mathrm{1}}{\sqrt{\mathrm{1}+\left({n}+\mathrm{1}\right)^{\mathrm{2}} }}\:\leqslant\:{argsh}\left({n}+\mathrm{1}\right)\:−{argsh}\left({n}\right)\leqslant\:\frac{\mathrm{1}}{\sqrt{\mathrm{1}+{n}^{\mathrm{2}} }}\:. \\ $$

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