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Question Number 65193 by mathmax by abdo last updated on 26/Jul/19
U_n is a sequence wich verify U_n +U_(n+1) =n for all integr n 1) calculate U_n intrem of n 2) find nature of the serie Σ (U_n /n^2 )
$${U}_{{n}} \:{is}\:{a}\:{sequence}\:{wich}\:{verify}\:{U}_{{n}} +{U}_{{n}+\mathrm{1}} ={n}\:{for}\:{all}\:{integr}\:{n} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{U}_{{n}} \:{intrem}\:{of}\:{n} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{nature}\:{of}\:{the}\:{serie}\:\Sigma\:\frac{{U}_{{n}} }{{n}^{\mathrm{2}} } \\ $$
Commented by mathmax by abdo last updated on 26/Jul/19
1) we have U_n +U_(n+1) =n ⇒Σ_(k=0) ^(n−1) (−1)^k (U_k +U_(k+1) ) =Σ_(k=0) ^(n−1) k(−1)^k ⇒U_0 +U_1 −U_1 −U_2 +...(−1)^(n−2) (U_(n−2) +U_(n−1) )+(−1)^(n−1) (U_(n−1) +U_n ) =Σ_(k=0) ^(n−1) k(−1)^k ⇒ U_0 +(−1)^(n−1) U_n =Σ_(k=0) ^(n−1) k(−1)^k ⇒(−1)^(n−1) U_n =Σ_(k=0) ^(n−1) k(−1)^k −U_0 ⇒U_n =Σ_(k=0) ^(n−1) k(−1)^(k+n−1) −(−1)^(n−1) U_0 ⇒ U_n =(−1)^(n−1) Σ_(k=0) ^(n−1) k(−1)^k +(−1)^n U_0 let p(x) =Σ_(k=0) ^(n−1) kx^k we have Σ_(k=0) ^(n−1) x^k =((1−x^n )/(1−x)) (x≠1) ⇒ Σ_(k=1) ^(n−1) kx^(k−1) =(((x^n −1)/(x−1)))^′ =((nx^(n−1) (x−1)−(x^n −1)×1)/((x−1)^2 )) =((nx^n −nx^(n−1) −x^n +1)/((x−1)^2 )) =(((n−1)x^n −nx^(n−1) +1)/((x−1)^2 )) ⇒ Σ_(k=1) ^(n−1) kx^k =(((n−1)x^(n+1) −nx^n +x)/((x−1)^2 )) ⇒ Σ_(k=0) ^(n−1) k(−1)^k =(((n−1)(−1)^(n+1) −n(−1)^n −1)/4) =((−n(−1)^n +(−1)^n −n(−1)^n −1)/4) =((−2n(−1)^n +(−1)^n −1)/4) ⇒ U_n =(((−1)^(n−1) )/4){ −2n(−1)^n +(−1)^n −1}+(−1)^n U_0 =−(1/4){−2n+1−(−1)^n }+(−1)^n U_0 ⇒ U_n =(1/4){2n−1 +(−1)^n } +(−1)^n U_0 2) Σ_(n=1) ^∞ (U_n /n^2 ) =Σ_(n=1) ^∞ (1/(2n)) −(1/4) Σ_(n=1) ^∞ (1/n^2 ) +(1/4)Σ_(n=1) ^∞ (((−1)^n )/n^2 ) +Σ_(n=1) ^∞ (((−1)^n U_0 )/n^2 ) the serie Σ(1/(2n)) diverges ⇒Σ(U_n /n^2 ) diverges...
$$\left.\mathrm{1}\right)\:{we}\:{have}\:{U}_{{n}} \:+{U}_{{n}+\mathrm{1}} ={n}\:\:\Rightarrow\sum_{{k}=\mathrm{0}} ^{{n}−\mathrm{1}} \left(−\mathrm{1}\right)^{{k}} \left({U}_{{k}} \:+{U}_{{k}+\mathrm{1}} \right)\:=\sum_{{k}=\mathrm{0}} ^{{n}−\mathrm{1}} {k}\left(−\mathrm{1}\right)^{{k}} \\ $$$$\Rightarrow{U}_{\mathrm{0}} +{U}_{\mathrm{1}} −{U}_{\mathrm{1}} −{U}_{\mathrm{2}} \:+…\left(−\mathrm{1}\right)^{{n}−\mathrm{2}} \left({U}_{{n}−\mathrm{2}} +{U}_{{n}−\mathrm{1}} \right)+\left(−\mathrm{1}\right)^{{n}−\mathrm{1}} \left({U}_{{n}−\mathrm{1}} +{U}_{{n}} \right) \\ $$$$=\sum_{{k}=\mathrm{0}} ^{{n}−\mathrm{1}} \:{k}\left(−\mathrm{1}\right)^{{k}} \:\Rightarrow \\ $$$${U}_{\mathrm{0}} \:\:+\left(−\mathrm{1}\right)^{{n}−\mathrm{1}} \:{U}_{{n}} =\sum_{{k}=\mathrm{0}} ^{{n}−\mathrm{1}} {k}\left(−\mathrm{1}\right)^{{k}} \:\Rightarrow\left(−\mathrm{1}\right)^{{n}−\mathrm{1}} \:{U}_{{n}} =\sum_{{k}=\mathrm{0}} ^{{n}−\mathrm{1}} {k}\left(−\mathrm{1}\right)^{{k}} \:−{U}_{\mathrm{0}} \\ $$$$\Rightarrow{U}_{{n}} =\sum_{{k}=\mathrm{0}} ^{{n}−\mathrm{1}} {k}\left(−\mathrm{1}\right)^{{k}+{n}−\mathrm{1}} \:−\left(−\mathrm{1}\right)^{{n}−\mathrm{1}} \:{U}_{\mathrm{0}} \:\Rightarrow \\ $$$${U}_{{n}} =\left(−\mathrm{1}\right)^{{n}−\mathrm{1}} \:\sum_{{k}=\mathrm{0}} ^{{n}−\mathrm{1}} {k}\left(−\mathrm{1}\right)^{{k}} \:+\left(−\mathrm{1}\right)^{{n}} \:{U}_{\mathrm{0}} \\ $$$${let}\:{p}\left({x}\right)\:=\sum_{{k}=\mathrm{0}} ^{{n}−\mathrm{1}} {kx}^{{k}} \:\:\:\:\:{we}\:{have}\:\sum_{{k}=\mathrm{0}} ^{{n}−\mathrm{1}} {x}^{{k}} \:=\frac{\mathrm{1}−{x}^{{n}} }{\mathrm{1}−{x}}\:\:\:\:\:\left({x}\neq\mathrm{1}\right)\:\Rightarrow \\ $$$$\sum_{{k}=\mathrm{1}} ^{{n}−\mathrm{1}} {kx}^{{k}−\mathrm{1}} \:=\left(\frac{{x}^{{n}} −\mathrm{1}}{{x}−\mathrm{1}}\right)^{'} \:=\frac{{nx}^{{n}−\mathrm{1}} \left({x}−\mathrm{1}\right)−\left({x}^{{n}} −\mathrm{1}\right)×\mathrm{1}}{\left({x}−\mathrm{1}\right)^{\mathrm{2}} } \\ $$$$=\frac{{nx}^{{n}} −{nx}^{{n}−\mathrm{1}} −{x}^{{n}} +\mathrm{1}}{\left({x}−\mathrm{1}\right)^{\mathrm{2}} }\:=\frac{\left({n}−\mathrm{1}\right){x}^{{n}} −{nx}^{{n}−\mathrm{1}} \:+\mathrm{1}}{\left({x}−\mathrm{1}\right)^{\mathrm{2}} }\:\Rightarrow \\ $$$$\sum_{{k}=\mathrm{1}} ^{{n}−\mathrm{1}} \:{kx}^{{k}} \:=\frac{\left({n}−\mathrm{1}\right){x}^{{n}+\mathrm{1}} −{nx}^{{n}} \:+{x}}{\left({x}−\mathrm{1}\right)^{\mathrm{2}} }\:\Rightarrow \\ $$$$\sum_{{k}=\mathrm{0}} ^{{n}−\mathrm{1}} \:{k}\left(−\mathrm{1}\right)^{{k}} \:=\frac{\left({n}−\mathrm{1}\right)\left(−\mathrm{1}\right)^{{n}+\mathrm{1}} −{n}\left(−\mathrm{1}\right)^{{n}} −\mathrm{1}}{\mathrm{4}} \\ $$$$=\frac{−{n}\left(−\mathrm{1}\right)^{{n}} \:+\left(−\mathrm{1}\right)^{{n}} −{n}\left(−\mathrm{1}\right)^{{n}} −\mathrm{1}}{\mathrm{4}}\:=\frac{−\mathrm{2}{n}\left(−\mathrm{1}\right)^{{n}} \:+\left(−\mathrm{1}\right)^{{n}} −\mathrm{1}}{\mathrm{4}}\:\Rightarrow \\ $$$${U}_{{n}} =\frac{\left(−\mathrm{1}\right)^{{n}−\mathrm{1}} }{\mathrm{4}}\left\{\:−\mathrm{2}{n}\left(−\mathrm{1}\right)^{{n}} +\left(−\mathrm{1}\right)^{{n}} −\mathrm{1}\right\}+\left(−\mathrm{1}\right)^{{n}} \:{U}_{\mathrm{0}} \\ $$$$=−\frac{\mathrm{1}}{\mathrm{4}}\left\{−\mathrm{2}{n}+\mathrm{1}−\left(−\mathrm{1}\right)^{{n}} \right\}+\left(−\mathrm{1}\right)^{{n}} \:{U}_{\mathrm{0}} \:\Rightarrow \\ $$$${U}_{{n}} =\frac{\mathrm{1}}{\mathrm{4}}\left\{\mathrm{2}{n}−\mathrm{1}\:+\left(−\mathrm{1}\right)^{{n}} \right\}\:+\left(−\mathrm{1}\right)^{{n}} \:{U}_{\mathrm{0}} \\ $$$$\left.\mathrm{2}\right)\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\frac{{U}_{{n}} }{{n}^{\mathrm{2}} }\:=\sum_{{n}=\mathrm{1}} ^{\infty} \frac{\mathrm{1}}{\mathrm{2}{n}}\:−\frac{\mathrm{1}}{\mathrm{4}}\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\frac{\mathrm{1}}{{n}^{\mathrm{2}} }\:+\frac{\mathrm{1}}{\mathrm{4}}\sum_{{n}=\mathrm{1}} ^{\infty} \:\frac{\left(−\mathrm{1}\right)^{{n}} }{{n}^{\mathrm{2}} }\:+\sum_{{n}=\mathrm{1}} ^{\infty} \frac{\left(−\mathrm{1}\right)^{{n}} {U}_{\mathrm{0}} }{{n}^{\mathrm{2}} } \\ $$$${the}\:{serie}\:\Sigma\frac{\mathrm{1}}{\mathrm{2}{n}}\:{diverges}\:\Rightarrow\Sigma\frac{{U}_{{n}} }{{n}^{\mathrm{2}} }\:{diverges}… \\ $$

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