Question Number 189394 by TUN last updated on 15/Mar/23
$${Prove}\:{that}: \\ $$$${ln}\left({n}+\mathrm{1}\right)<\frac{\mathrm{1}}{\:\sqrt{\mathrm{1}^{\mathrm{2}} +\mathrm{1}}}+\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}^{\mathrm{2}} +\mathrm{2}}}+...+\frac{\mathrm{1}}{\:\sqrt{{n}^{\mathrm{2}} +{n}}}\left(\forall{n}\in{N}^{\ast} \right) \\ $$