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let-U-n-1-2-2-2-3-2-n-2-1-4-2-4-3-4-n-4-1-find-lim-n-U-n-2-calculate-n-1-U-n-




Question Number 58354 by maxmathsup by imad last updated on 21/Apr/19
let U_n =((1^2  +2^2  +3^2  +....+n^2 )/(1^4  +2^4  +3^4  +....+n^4 ))  1)find lim_(n→+∞) U_n   2) calculate Σ_(n=1) ^∞  U_n
$${let}\:{U}_{{n}} =\frac{\mathrm{1}^{\mathrm{2}} \:+\mathrm{2}^{\mathrm{2}} \:+\mathrm{3}^{\mathrm{2}} \:+….+{n}^{\mathrm{2}} }{\mathrm{1}^{\mathrm{4}} \:+\mathrm{2}^{\mathrm{4}} \:+\mathrm{3}^{\mathrm{4}} \:+….+{n}^{\mathrm{4}} } \\ $$$$\left.\mathrm{1}\right){find}\:{lim}_{{n}\rightarrow+\infty} {U}_{{n}} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\sum_{{n}=\mathrm{1}} ^{\infty} \:{U}_{{n}} \\ $$
Answered by tanmay last updated on 22/Apr/19
U_n   here =(N_r /D_r )  N_r =Σ_(r=1) ^n r^2 =(n^3 /3)+(n^2 /2)+(n/6)  D_r =Σ_(r=1) ^n r^4 =(n^5 /5)+(n^4 /2)+(n^3 /3)−(n/(30))  each term of N_r < each term of D_r   lim_(n→∞)  (((n^3 /3)+(n^2 /2)+(n/6))/((n^5 /5)+(n^4 /2)+(n^3 /3)−(n/(30))))  devide N_r  and D_r  by  n^5 →  lim_(n→∞)  (((1/(3n^2 ))+(1/(2n^3 ))+(1/(6n^4 )))/((1/5)+(1/(2n))+(1/(3n^2 ))−(1/(30n^4 ))))  =((0+0+0)/((1/5)+0+0−0))  =0
$${U}_{{n}} \:\:{here}\:=\frac{{N}_{{r}} }{{D}_{{r}} } \\ $$$${N}_{{r}} =\underset{{r}=\mathrm{1}} {\overset{{n}} {\sum}}{r}^{\mathrm{2}} =\frac{{n}^{\mathrm{3}} }{\mathrm{3}}+\frac{{n}^{\mathrm{2}} }{\mathrm{2}}+\frac{{n}}{\mathrm{6}} \\ $$$${D}_{{r}} =\underset{{r}=\mathrm{1}} {\overset{{n}} {\sum}}{r}^{\mathrm{4}} =\frac{{n}^{\mathrm{5}} }{\mathrm{5}}+\frac{{n}^{\mathrm{4}} }{\mathrm{2}}+\frac{{n}^{\mathrm{3}} }{\mathrm{3}}−\frac{{n}}{\mathrm{30}} \\ $$$${each}\:{term}\:{of}\:{N}_{{r}} <\:{each}\:{term}\:{of}\:{D}_{{r}} \\ $$$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\frac{\frac{{n}^{\mathrm{3}} }{\mathrm{3}}+\frac{{n}^{\mathrm{2}} }{\mathrm{2}}+\frac{{n}}{\mathrm{6}}}{\frac{{n}^{\mathrm{5}} }{\mathrm{5}}+\frac{{n}^{\mathrm{4}} }{\mathrm{2}}+\frac{{n}^{\mathrm{3}} }{\mathrm{3}}−\frac{{n}}{\mathrm{30}}} \\ $$$${devide}\:{N}_{{r}} \:{and}\:{D}_{{r}} \:{by}\:\:{n}^{\mathrm{5}} \rightarrow \\ $$$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\frac{\frac{\mathrm{1}}{\mathrm{3}{n}^{\mathrm{2}} }+\frac{\mathrm{1}}{\mathrm{2}{n}^{\mathrm{3}} }+\frac{\mathrm{1}}{\mathrm{6}{n}^{\mathrm{4}} }}{\frac{\mathrm{1}}{\mathrm{5}}+\frac{\mathrm{1}}{\mathrm{2}{n}}+\frac{\mathrm{1}}{\mathrm{3}{n}^{\mathrm{2}} }−\frac{\mathrm{1}}{\mathrm{30}{n}^{\mathrm{4}} }} \\ $$$$=\frac{\mathrm{0}+\mathrm{0}+\mathrm{0}}{\frac{\mathrm{1}}{\mathrm{5}}+\mathrm{0}+\mathrm{0}−\mathrm{0}} \\ $$$$=\mathrm{0} \\ $$$$ \\ $$$$ \\ $$$$ \\ $$

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