Question Number 144107 by SOMEDAVONG last updated on 21/Jun/21
$$\mathrm{A}=\underset{\mathrm{n}\rightarrow+\propto} {\mathrm{lim}}\frac{\mathrm{1}+\sqrt[{\mathrm{7}}]{\mathrm{2}}+\sqrt[{\mathrm{7}}]{\mathrm{3}}+\sqrt[{\mathrm{7}}]{\mathrm{4}}+.....+\sqrt[{\mathrm{7}}]{\mathrm{n}}}{\:\sqrt[{\mathrm{7}}]{\mathrm{n}^{\mathrm{9}} }}\:=? \\ $$
Answered by Dwaipayan Shikari last updated on 21/Jun/21
$${A}=\underset{{n}\rightarrow\infty} {\mathrm{lim}}\frac{\mathrm{1}+\sqrt[{\mathrm{7}}]{\mathrm{2}}+..+\sqrt[{\mathrm{7}}]{{n}}}{\:\sqrt[{\mathrm{7}}]{{n}^{\mathrm{8}} }} \\ $$$$=\frac{\mathrm{1}}{{n}}\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}\sqrt[{\mathrm{7}}]{\frac{{k}}{{n}}}=\int_{\mathrm{0}} ^{\mathrm{1}} \sqrt[{\mathrm{7}}]{{x}}\:{dx}=\frac{\mathrm{7}}{\mathrm{8}} \\ $$