Question Number 80614 by M±th+et£s last updated on 04/Feb/20
Commented by mathmax by abdo last updated on 04/Feb/20
$${A}_{{n}} =\frac{\mathrm{1}}{{n}}\sum_{{k}=\mathrm{1}} ^{{n}} {ln}\left({a}+\frac{{k}}{{n}}\right)\Rightarrow{A}_{{n}} \:{is}\:{a}\:{Rieman}\:{sum} \\ $$$${and}\:{lim}_{{n}\rightarrow+\infty} \:{A}_{{n}} =\int_{\mathrm{0}} ^{\mathrm{1}} {ln}\left({a}+{x}\right){dx}\:=_{{a}+{x}={t}} \:\:\int_{{a}} ^{\mathrm{1}+{a}} {ln}\left({t}\right){dt} \\ $$$$=\left[{tln}\left({t}\right)−{t}\right]_{{a}} ^{\mathrm{1}+{a}} \:=\left(\mathrm{1}+{a}\right){ln}\left(\mathrm{1}+{a}\right)−\left(\mathrm{1}+{a}\right)−\left({alna}−{a}\right) \\ $$$$=\left(\mathrm{1}+{a}\right){ln}\left(\mathrm{1}+{a}\right)−\mathrm{1}−{a}−{alna}+{a} \\ $$$$=\left(\mathrm{1}+{a}\right){ln}\left(\mathrm{1}+{a}\right)−{alna}\:−\mathrm{1} \\ $$