Question Number 61907 by naka3546 last updated on 11/Jun/19
Commented by MJS last updated on 12/Jun/19
$$\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{{n}}=\infty\:\Rightarrow\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{\sqrt[{{k}}]{{n}}}=\infty\:\mathrm{for}\:{k}\in\mathbb{N}^{\bigstar} \\ $$$$ \\ $$$$\mathrm{calculating}\:\Omega_{{n}} \:\mathrm{I}\:\mathrm{get}: \\ $$$$\Omega_{\mathrm{125}} \approx.\mathrm{849696} \\ $$$$\Omega_{\mathrm{250}} \approx.\mathrm{846055} \\ $$$$\Omega_{\mathrm{500}} \approx.\mathrm{843624} \\ $$$$\Omega_{\mathrm{1000}} \approx.\mathrm{842011} \\ $$$$\Omega_{\mathrm{2000}} \approx.\mathrm{840945} \\ $$
Answered by tanmay last updated on 11/Jun/19
![lim_(n→∞) (N_r /D_r ) N_r =[((1/1))^(1/5) +((1/2))^(1/5) +((1/3))^(1/5) +...+((1/n))^(1/5) ]^(2/3) D_r =[((1/1))^(1/3) +((1/2))^(1/3) +((1/3))^(1/3) +...+((1/n))^(1/3) ]^(4/5) N_r =[((1/1))^(1/5) +((1/2))^(1/5) +((1/3))^(1/5) +...+((1/n))^(1/5) ]^((10)/(15)) D_r =[((1/1))^(1/3) +((1/2))^(1/3) +((1/3))^(1/3) +...+((1/n))^(1/3) ]^((12)/(15)) N_r =[{((1/1))^(1/5) +((1/2))^(1/5) +((1/3))^(1/5) +...+((1/n))^(1/5) }^5 ]^(2/(15)) =Σ_(n=1) ^n [{((1/n))^(1/5) }^5 ]^(2/(15)) and D_r =Σ_(n=1) ^∞ [{((1/n))^(1/3) }^6 ]^(2/(15)) N_r =[(1/1)+(1/2)+..+(1/n)+g(n)]^(2/(15)) D_r =[(1/1^2 )+(1/2^2 )+..+(1/n^2 )+h(n)]^(2/(15)) (1/n^2 )<(1/n) D_r <N_r wait...](Q61920.png)
$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\frac{{N}_{{r}} }{{D}_{{r}} } \\ $$$${N}_{{r}} =\left[\left(\frac{\mathrm{1}}{\mathrm{1}}\right)^{\frac{\mathrm{1}}{\mathrm{5}}} +\left(\frac{\mathrm{1}}{\mathrm{2}}\right)^{\frac{\mathrm{1}}{\mathrm{5}}} +\left(\frac{\mathrm{1}}{\mathrm{3}}\right)^{\frac{\mathrm{1}}{\mathrm{5}}} +...+\left(\frac{\mathrm{1}}{{n}}\right)^{\frac{\mathrm{1}}{\mathrm{5}}} \right]^{\frac{\mathrm{2}}{\mathrm{3}}} \\ $$$${D}_{{r}} =\left[\left(\frac{\mathrm{1}}{\mathrm{1}}\right)^{\frac{\mathrm{1}}{\mathrm{3}}} +\left(\frac{\mathrm{1}}{\mathrm{2}}\right)^{\frac{\mathrm{1}}{\mathrm{3}}} +\left(\frac{\mathrm{1}}{\mathrm{3}}\right)^{\frac{\mathrm{1}}{\mathrm{3}}} +...+\left(\frac{\mathrm{1}}{{n}}\right)^{\frac{\mathrm{1}}{\mathrm{3}}} \right]^{\frac{\mathrm{4}}{\mathrm{5}}} \\ $$$${N}_{{r}} =\left[\left(\frac{\mathrm{1}}{\mathrm{1}}\right)^{\frac{\mathrm{1}}{\mathrm{5}}} +\left(\frac{\mathrm{1}}{\mathrm{2}}\right)^{\frac{\mathrm{1}}{\mathrm{5}}} +\left(\frac{\mathrm{1}}{\mathrm{3}}\right)^{\frac{\mathrm{1}}{\mathrm{5}}} +...+\left(\frac{\mathrm{1}}{{n}}\right)^{\frac{\mathrm{1}}{\mathrm{5}}} \right]^{\frac{\mathrm{10}}{\mathrm{15}}} \\ $$$${D}_{{r}} =\left[\left(\frac{\mathrm{1}}{\mathrm{1}}\right)^{\frac{\mathrm{1}}{\mathrm{3}}} +\left(\frac{\mathrm{1}}{\mathrm{2}}\right)^{\frac{\mathrm{1}}{\mathrm{3}}} +\left(\frac{\mathrm{1}}{\mathrm{3}}\right)^{\frac{\mathrm{1}}{\mathrm{3}}} +...+\left(\frac{\mathrm{1}}{{n}}\right)^{\frac{\mathrm{1}}{\mathrm{3}}} \right]^{\frac{\mathrm{12}}{\mathrm{15}}} \\ $$$${N}_{{r}} =\left[\left\{\left(\frac{\mathrm{1}}{\mathrm{1}}\right)^{\frac{\mathrm{1}}{\mathrm{5}}} +\left(\frac{\mathrm{1}}{\mathrm{2}}\right)^{\frac{\mathrm{1}}{\mathrm{5}}} +\left(\frac{\mathrm{1}}{\mathrm{3}}\right)^{\frac{\mathrm{1}}{\mathrm{5}}} +...+\left(\frac{\mathrm{1}}{{n}}\right)^{\frac{\mathrm{1}}{\mathrm{5}}} \right\}^{\mathrm{5}} \right]^{\frac{\mathrm{2}}{\mathrm{15}}} \\ $$$$=\underset{{n}=\mathrm{1}} {\overset{{n}} {\sum}}\left[\left\{\left(\frac{\mathrm{1}}{{n}}\right)^{\frac{\mathrm{1}}{\mathrm{5}}} \right\}^{\mathrm{5}} \right]^{\frac{\mathrm{2}}{\mathrm{15}}} \:{and}\:{D}_{{r}} =\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\left[\left\{\left(\frac{\mathrm{1}}{{n}}\right)^{\frac{\mathrm{1}}{\mathrm{3}}} \right\}^{\mathrm{6}} \right]^{\frac{\mathrm{2}}{\mathrm{15}}} \\ $$$${N}_{{r}} =\left[\frac{\mathrm{1}}{\mathrm{1}}+\frac{\mathrm{1}}{\mathrm{2}}+..+\frac{\mathrm{1}}{{n}}+{g}\left({n}\right)\right]^{\frac{\mathrm{2}}{\mathrm{15}}} \\ $$$${D}_{{r}} =\left[\frac{\mathrm{1}}{\mathrm{1}^{\mathrm{2}} }+\frac{\mathrm{1}}{\mathrm{2}^{\mathrm{2}} }+..+\frac{\mathrm{1}}{{n}^{\mathrm{2}} }+{h}\left({n}\right)\right]^{\frac{\mathrm{2}}{\mathrm{15}}} \\ $$$$\frac{\mathrm{1}}{{n}^{\mathrm{2}} }<\frac{\mathrm{1}}{{n}}\:\:\:\:{D}_{{r}} <{N}_{{r}} \\ $$$${wait}... \\ $$$$ \\ $$