Question Number 30174 by abdo imad last updated on 18/Feb/18
$${let}\:\:{u}_{{n}} =\:\sum_{{k}=\mathrm{1}} ^{{n}} \:\frac{\mathrm{1}}{{k}} \\ $$$$\mathrm{1}.\:{prove}\:{that}\:{ln}\left({n}+\mathrm{1}\right)\leqslant{u}_{{n}} \leqslant{ln}\left({n}\right)\:+\mathrm{1} \\ $$$$\mathrm{2}.\:{show}\:{that}\:{u}_{{n}} \:\:_{{n}\rightarrow\infty} \sim\:{ln}\left({n}\right)\:\:. \\ $$