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Question Number 61907 by naka3546 last updated on 11/Jun/19

Commented by MJS last updated on 12/Jun/19

$\underset{n = 1}{\sum^{\infty}} \frac{1}{n} = \infty \Rightarrow \underset{n = 1}{\sum^{\infty}} \frac{1}{\sqrt[k]{n}} = \infty for k \in N^{★}$ $calculating Ω_{n} I get :$ $Ω_{125} \approx . 849696$ $Ω_{250} \approx . 846055$ $Ω_{500} \approx . 843624$ $Ω_{1000} \approx . 842011$ $Ω_{2000} \approx . 840945$
	Answered by tanmay last updated on 11/Jun/19
	$lim_(n→∞) (N_r /D_r ) N_r =[((1/1))^(1/5) +((1/2))^(1/5) +((1/3))^(1/5) +...+((1/n))^(1/5) ]^(2/3) D_r =[((1/1))^(1/3) +((1/2))^(1/3) +((1/3))^(1/3) +...+((1/n))^(1/3) ]^(4/5) N_r =[((1/1))^(1/5) +((1/2))^(1/5) +((1/3))^(1/5) +...+((1/n))^(1/5) ]^((10)/(15)) D_r =[((1/1))^(1/3) +((1/2))^(1/3) +((1/3))^(1/3) +...+((1/n))^(1/3) ]^((12)/(15)) N_r =[{((1/1))^(1/5) +((1/2))^(1/5) +((1/3))^(1/5) +...+((1/n))^(1/5) }^5 ]^(2/(15)) =Σ_(n=1) ^n [{((1/n))^(1/5) }^5 ]^(2/(15)) and D_r =Σ_(n=1) ^∞ [{((1/n))^(1/3) }^6 ]^(2/(15)) N_r =[(1/1)+(1/2)+..+(1/n)+g(n)]^(2/(15)) D_r =[(1/1^2 )+(1/2^2 )+..+(1/n^2 )+h(n)]^(2/(15)) (1/n^2 )<(1/n) D_r <N_r wait...$
	$\lim_{n \to \infty} \frac{N_{r}}{D_{r}}$ $N_{r} = {[{(\frac{1}{1})}^{\frac{1}{5}} + {(\frac{1}{2})}^{\frac{1}{5}} + {(\frac{1}{3})}^{\frac{1}{5}} + . . . + {(\frac{1}{n})}^{\frac{1}{5}}]}^{\frac{2}{3}}$ $D_{r} = {[{(\frac{1}{1})}^{\frac{1}{3}} + {(\frac{1}{2})}^{\frac{1}{3}} + {(\frac{1}{3})}^{\frac{1}{3}} + . . . + {(\frac{1}{n})}^{\frac{1}{3}}]}^{\frac{4}{5}}$ $N_{r} = {[{(\frac{1}{1})}^{\frac{1}{5}} + {(\frac{1}{2})}^{\frac{1}{5}} + {(\frac{1}{3})}^{\frac{1}{5}} + . . . + {(\frac{1}{n})}^{\frac{1}{5}}]}^{\frac{10}{15}}$ $D_{r} = {[{(\frac{1}{1})}^{\frac{1}{3}} + {(\frac{1}{2})}^{\frac{1}{3}} + {(\frac{1}{3})}^{\frac{1}{3}} + . . . + {(\frac{1}{n})}^{\frac{1}{3}}]}^{\frac{12}{15}}$ $N_{r} = {[{{(\frac{1}{1})}^{\frac{1}{5}} + {(\frac{1}{2})}^{\frac{1}{5}} + {(\frac{1}{3})}^{\frac{1}{5}} + . . . + {(\frac{1}{n})}^{\frac{1}{5}}}^{5}]}^{\frac{2}{15}}$ $= \underset{n = 1}{\sum^{n}} {[{{(\frac{1}{n})}^{\frac{1}{5}}}^{5}]}^{\frac{2}{15}} a n d D_{r} = \underset{n = 1}{\sum^{\infty}} {[{{(\frac{1}{n})}^{\frac{1}{3}}}^{6}]}^{\frac{2}{15}}$ $N_{r} = {[\frac{1}{1} + \frac{1}{2} + . . + \frac{1}{n} + g (n)]}^{\frac{2}{15}}$ $D_{r} = {[\frac{1}{1^{2}} + \frac{1}{2^{2}} + . . + \frac{1}{n^{2}} + h (n)]}^{\frac{2}{15}}$ $\frac{1}{n^{2}} < \frac{1}{n} D_{r} < N_{r}$ $w a i t . . .$
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