Question Number 107080 by Khalmohmmad last updated on 08/Aug/20
Answered by abdomsup last updated on 08/Aug/20
$${A}_{{n}} =\frac{{n}^{{n}} \left\{\frac{{n}}{{n}^{{n}} }+\frac{{n}^{\mathrm{2}} }{{n}^{{n}} }+....+\mathrm{1}\right\}}{{n}^{{n}} \left\{\frac{\mathrm{1}}{{n}^{{n}} }\:+\left(\frac{\mathrm{2}}{{n}}\right)^{{n}\:} +\left(\frac{\mathrm{3}}{{n}^{} }\right)^{{n}} +....\mathrm{1}\right\}} \\ $$$${A}_{{n}} =\frac{\frac{\mathrm{1}}{{n}^{{n}−\mathrm{1}} }+\frac{\mathrm{1}}{{n}^{{n}−\mathrm{2}} }+....+\mathrm{1}}{\left(\frac{\mathrm{1}}{{n}}\right)^{{n}} \:+\left(\frac{\mathrm{2}}{{n}}\right)^{{n}} +...+\mathrm{1}}\:\Rightarrow \\ $$$${lim}_{{n}\rightarrow+\infty} \:{A}_{{n}} =\mathrm{1} \\ $$