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Question Number 61232 by maxmathsup by imad last updated on 30/May/19
let U_n =∫_1 ^(+∞)    (([nx]−[(n−1)x])/x^3 ) dx  with n≥1  1) find U_n  interms of n  2) find lim_(n→+∞)  U_n   3) study the serie Σ_(n=1) ^∞  U_n
$${let}\:{U}_{{n}} =\int_{\mathrm{1}} ^{+\infty} \:\:\:\frac{\left[{nx}\right]−\left[\left({n}−\mathrm{1}\right){x}\right]}{{x}^{\mathrm{3}} }\:{dx}\:\:{with}\:{n}\geqslant\mathrm{1} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{U}_{{n}} \:{interms}\:{of}\:{n} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{lim}_{{n}\rightarrow+\infty} \:{U}_{{n}} \\ $$$$\left.\mathrm{3}\right)\:{study}\:{the}\:{serie}\:\sum_{{n}=\mathrm{1}} ^{\infty} \:{U}_{{n}} \\ $$
Commented by maxmathsup by imad last updated on 01/Jun/19
1) we have U_n =∫_1 ^(+∞)   (([nx])/x^3 )dx −∫_1 ^(+∞)   (([(n−1)x])/x^3 ) dx   ∫_1 ^(+∞)    (([nx])/x^3 ) dx =_(nx=t)      n^3 ∫_n ^(+∞)   (([t])/t^3 ) (dt/n) =n^2  ∫_n ^(+∞)   (([t])/t^3 ) dt  =n^2  Σ_(k=n) ^(+∞)   ∫_k ^(k+1)   (k/t^3 ) dt =n^2  Σ_(k=n) ^∞  k ∫_k ^(k+1)  t^(−3) dt =n^2 Σ_(k=n) ^∞ k[−(1/2) t^(−2) ]_k ^(k+1)   =−(n^2 /2) Σ_(k=n) ^∞  k{  (1/((k+1)^2 )) −(1/k^2 )} =−(n^2 /2){ Σ_(k=n) ^∞   (k/((k+1)^2 )) −Σ_(k=n) ^∞  (1/k)}  we have  Σ_(k=n) ^∞  (k/((k+1)^2 )) =Σ_(k=n+1) ^∞   ((k−1)/k^2 ) =Σ_(k=n+1) ^∞  (1/k) −Σ_(k=n+1) ^∞  (1/k^2 ) ⇒  ∫_1 ^(+∞)   (([nx])/x^3 ) dx =−(n^2 /2){ Σ_(k=n) ^∞  (1/k) −(1/(n+1)) −Σ_(k=n+1) ^∞  (1/k^2 ) −Σ_(k=n) ^∞  (1/k)}  =(n^2 /2) Σ_(k=n+1) ^∞  (1/k^2 ) + (n^2 /(2(n+1)))  ∫_1 ^∞    (([(n−1)x])/x^3 ) dx =_((n−1)x =t)   (n−1)^3     ∫_(n−1) ^(+∞)    (([t])/t^3 ) (dt/(n−1))  =(n−1)^2  ∫_(n−1) ^(+∞)   (([t])/t^3 ) dt =(n−1)^2  Σ_(k=n−1) ^∞  ∫_k ^(k+1)  (k/t^3 ) dt  =(n−1)^2  Σ_(k=n−1) ^∞  k ∫_k ^(k+1)   (dt/t^(−3) ) =(n−1)^2  Σ_(k=n−1) ^∞  k[−(1/(2t^2 ))]_m ^(k+1)   =−(((n−1)^2 )/2) Σ_(k=n−1) ^∞  k{ (1/((k+1)^2 )) −(1/k^2 )} =−(((n−1)^2 )/2){ Σ_(k=n−1) ^∞  (k/((k+1)^2 )) −Σ_(k=n−1) ^∞  (1/k)}  Σ_(k=n−1) ^∞  (k/((k+1)^2 ))  =Σ_(k=n) ^∞   ((k−1)/k^2 ) =Σ_(k=n) ^∞  (1/k) −Σ_(k=n) ^∞  (1/k^2 )  ⇒  ∫_1 ^∞   (([(n−1)x])/x^3 ) dx =−(((n−1)^2 )/2){ Σ_(k=n) ^∞  (1/k) −Σ_(k=n) ^∞  (1/k^2 ) −(1/(n−1)) −Σ_(k=n) ^∞  (1/k)}  =−(((n−1)^2 )/2){−Σ_(k=n) ^∞  (1/k^2 ) −(1/(n−1)) } =(((n−1)^2 )/2) Σ_(k=n) ^∞  (1/k^2 ) +((n−1)/2) ⇒  U_n =(n^2 /2) Σ_(k=n+1) ^∞  (1/k^2 ) +(n^2 /(2(n+1))) −(((n−1)^2 )/2) Σ_(k=n) ^∞  (1/k^2 ) −((n−1)/2) .
$$\left.\mathrm{1}\right)\:{we}\:{have}\:{U}_{{n}} =\int_{\mathrm{1}} ^{+\infty} \:\:\frac{\left[{nx}\right]}{{x}^{\mathrm{3}} }{dx}\:−\int_{\mathrm{1}} ^{+\infty} \:\:\frac{\left[\left({n}−\mathrm{1}\right){x}\right]}{{x}^{\mathrm{3}} }\:{dx}\: \\ $$$$\int_{\mathrm{1}} ^{+\infty} \:\:\:\frac{\left[{nx}\right]}{{x}^{\mathrm{3}} }\:{dx}\:=_{{nx}={t}} \:\:\:\:\:{n}^{\mathrm{3}} \int_{{n}} ^{+\infty} \:\:\frac{\left[{t}\right]}{{t}^{\mathrm{3}} }\:\frac{{dt}}{{n}}\:={n}^{\mathrm{2}} \:\int_{{n}} ^{+\infty} \:\:\frac{\left[{t}\right]}{{t}^{\mathrm{3}} }\:{dt} \\ $$$$={n}^{\mathrm{2}} \:\sum_{{k}={n}} ^{+\infty} \:\:\int_{{k}} ^{{k}+\mathrm{1}} \:\:\frac{{k}}{{t}^{\mathrm{3}} }\:{dt}\:={n}^{\mathrm{2}} \:\sum_{{k}={n}} ^{\infty} \:{k}\:\int_{{k}} ^{{k}+\mathrm{1}} \:{t}^{−\mathrm{3}} {dt}\:={n}^{\mathrm{2}} \sum_{{k}={n}} ^{\infty} {k}\left[−\frac{\mathrm{1}}{\mathrm{2}}\:{t}^{−\mathrm{2}} \right]_{{k}} ^{{k}+\mathrm{1}} \\ $$$$=−\frac{{n}^{\mathrm{2}} }{\mathrm{2}}\:\sum_{{k}={n}} ^{\infty} \:{k}\left\{\:\:\frac{\mathrm{1}}{\left({k}+\mathrm{1}\right)^{\mathrm{2}} }\:−\frac{\mathrm{1}}{{k}^{\mathrm{2}} }\right\}\:=−\frac{{n}^{\mathrm{2}} }{\mathrm{2}}\left\{\:\sum_{{k}={n}} ^{\infty} \:\:\frac{{k}}{\left({k}+\mathrm{1}\right)^{\mathrm{2}} }\:−\sum_{{k}={n}} ^{\infty} \:\frac{\mathrm{1}}{{k}}\right\} \\ $$$${we}\:{have}\:\:\sum_{{k}={n}} ^{\infty} \:\frac{{k}}{\left({k}+\mathrm{1}\right)^{\mathrm{2}} }\:=\sum_{{k}={n}+\mathrm{1}} ^{\infty} \:\:\frac{{k}−\mathrm{1}}{{k}^{\mathrm{2}} }\:=\sum_{{k}={n}+\mathrm{1}} ^{\infty} \:\frac{\mathrm{1}}{{k}}\:−\sum_{{k}={n}+\mathrm{1}} ^{\infty} \:\frac{\mathrm{1}}{{k}^{\mathrm{2}} }\:\Rightarrow \\ $$$$\int_{\mathrm{1}} ^{+\infty} \:\:\frac{\left[{nx}\right]}{{x}^{\mathrm{3}} }\:{dx}\:=−\frac{{n}^{\mathrm{2}} }{\mathrm{2}}\left\{\:\sum_{{k}={n}} ^{\infty} \:\frac{\mathrm{1}}{{k}}\:−\frac{\mathrm{1}}{{n}+\mathrm{1}}\:−\sum_{{k}={n}+\mathrm{1}} ^{\infty} \:\frac{\mathrm{1}}{{k}^{\mathrm{2}} }\:−\sum_{{k}={n}} ^{\infty} \:\frac{\mathrm{1}}{{k}}\right\} \\ $$$$=\frac{{n}^{\mathrm{2}} }{\mathrm{2}}\:\sum_{{k}={n}+\mathrm{1}} ^{\infty} \:\frac{\mathrm{1}}{{k}^{\mathrm{2}} }\:+\:\frac{{n}^{\mathrm{2}} }{\mathrm{2}\left({n}+\mathrm{1}\right)} \\ $$$$\int_{\mathrm{1}} ^{\infty} \:\:\:\frac{\left[\left({n}−\mathrm{1}\right){x}\right]}{{x}^{\mathrm{3}} }\:{dx}\:=_{\left({n}−\mathrm{1}\right){x}\:={t}} \:\:\left({n}−\mathrm{1}\right)^{\mathrm{3}} \:\:\:\:\int_{{n}−\mathrm{1}} ^{+\infty} \:\:\:\frac{\left[{t}\right]}{{t}^{\mathrm{3}} }\:\frac{{dt}}{{n}−\mathrm{1}} \\ $$$$=\left({n}−\mathrm{1}\right)^{\mathrm{2}} \:\int_{{n}−\mathrm{1}} ^{+\infty} \:\:\frac{\left[{t}\right]}{{t}^{\mathrm{3}} }\:{dt}\:=\left({n}−\mathrm{1}\right)^{\mathrm{2}} \:\sum_{{k}={n}−\mathrm{1}} ^{\infty} \:\int_{{k}} ^{{k}+\mathrm{1}} \:\frac{{k}}{{t}^{\mathrm{3}} }\:{dt} \\ $$$$=\left({n}−\mathrm{1}\right)^{\mathrm{2}} \:\sum_{{k}={n}−\mathrm{1}} ^{\infty} \:{k}\:\int_{{k}} ^{{k}+\mathrm{1}} \:\:\frac{{dt}}{{t}^{−\mathrm{3}} }\:=\left({n}−\mathrm{1}\right)^{\mathrm{2}} \:\sum_{{k}={n}−\mathrm{1}} ^{\infty} \:{k}\left[−\frac{\mathrm{1}}{\mathrm{2}{t}^{\mathrm{2}} }\right]_{{m}} ^{{k}+\mathrm{1}} \\ $$$$=−\frac{\left({n}−\mathrm{1}\right)^{\mathrm{2}} }{\mathrm{2}}\:\sum_{{k}={n}−\mathrm{1}} ^{\infty} \:{k}\left\{\:\frac{\mathrm{1}}{\left({k}+\mathrm{1}\right)^{\mathrm{2}} }\:−\frac{\mathrm{1}}{{k}^{\mathrm{2}} }\right\}\:=−\frac{\left({n}−\mathrm{1}\right)^{\mathrm{2}} }{\mathrm{2}}\left\{\:\sum_{{k}={n}−\mathrm{1}} ^{\infty} \:\frac{{k}}{\left({k}+\mathrm{1}\right)^{\mathrm{2}} }\:−\sum_{{k}={n}−\mathrm{1}} ^{\infty} \:\frac{\mathrm{1}}{{k}}\right\} \\ $$$$\sum_{{k}={n}−\mathrm{1}} ^{\infty} \:\frac{{k}}{\left({k}+\mathrm{1}\right)^{\mathrm{2}} }\:\:=\sum_{{k}={n}} ^{\infty} \:\:\frac{{k}−\mathrm{1}}{{k}^{\mathrm{2}} }\:=\sum_{{k}={n}} ^{\infty} \:\frac{\mathrm{1}}{{k}}\:−\sum_{{k}={n}} ^{\infty} \:\frac{\mathrm{1}}{{k}^{\mathrm{2}} }\:\:\Rightarrow \\ $$$$\int_{\mathrm{1}} ^{\infty} \:\:\frac{\left[\left({n}−\mathrm{1}\right){x}\right]}{{x}^{\mathrm{3}} }\:{dx}\:=−\frac{\left({n}−\mathrm{1}\right)^{\mathrm{2}} }{\mathrm{2}}\left\{\:\sum_{{k}={n}} ^{\infty} \:\frac{\mathrm{1}}{{k}}\:−\sum_{{k}={n}} ^{\infty} \:\frac{\mathrm{1}}{{k}^{\mathrm{2}} }\:−\frac{\mathrm{1}}{{n}−\mathrm{1}}\:−\sum_{{k}={n}} ^{\infty} \:\frac{\mathrm{1}}{{k}}\right\} \\ $$$$=−\frac{\left({n}−\mathrm{1}\right)^{\mathrm{2}} }{\mathrm{2}}\left\{−\sum_{{k}={n}} ^{\infty} \:\frac{\mathrm{1}}{{k}^{\mathrm{2}} }\:−\frac{\mathrm{1}}{{n}−\mathrm{1}}\:\right\}\:=\frac{\left({n}−\mathrm{1}\right)^{\mathrm{2}} }{\mathrm{2}}\:\sum_{{k}={n}} ^{\infty} \:\frac{\mathrm{1}}{{k}^{\mathrm{2}} }\:+\frac{{n}−\mathrm{1}}{\mathrm{2}}\:\Rightarrow \\ $$$${U}_{{n}} =\frac{{n}^{\mathrm{2}} }{\mathrm{2}}\:\sum_{{k}={n}+\mathrm{1}} ^{\infty} \:\frac{\mathrm{1}}{{k}^{\mathrm{2}} }\:+\frac{{n}^{\mathrm{2}} }{\mathrm{2}\left({n}+\mathrm{1}\right)}\:−\frac{\left({n}−\mathrm{1}\right)^{\mathrm{2}} }{\mathrm{2}}\:\sum_{{k}={n}} ^{\infty} \:\frac{\mathrm{1}}{{k}^{\mathrm{2}} }\:−\frac{{n}−\mathrm{1}}{\mathrm{2}}\:. \\ $$$$ \\ $$$$ \\ $$
Commented by maxmathsup by imad last updated on 02/Jun/19
we have ξ(2) =Σ_(k=1) ^∞  (1/k^2 ) =Σ_(k=1) ^n  (1/k^2 ) +Σ_(k=n+1) ^∞  (1/k^2 ) =ξ_n (2) +Σ_(k=n+1) ^∞  (1/k^2 ) ⇒  Σ_(k=n+1) ^∞  (1/k^2 ) =ξ(2)−ξ_n (2)  also  Σ_(k=n) ^∞  (1/k^2 ) =ξ(2)−ξ_(n−1) (2) ⇒  U_n =(n^2 /2)( ξ(2)−ξ_n (2))−(((n−1)^2 )/2) (ξ(2)−ξ_(n−1) (2)) +(n^2 /(2(n+1))) −((n−1)/2)  =((n^2 −(n−1)^2 )/2)ξ(2) −(n^2 /2)ξ_n (2)−(((n−1)^2 )/2) ξ_(n−1) (2)  +((n^2 −(n−1)(n+1))/(2(n+1)))  =((2n−1)/2) ξ(2) −(n^2 /2)( ξ_(n−1) (2) +(1/n^2 ))−(((n−1)^2 )/2) ξ_(n−1) (2)  + (1/(2(n+1)))  =(n−(1/2))ξ(2) −((n^2  +n^2 −2n+1)/2) ξ_(n−1) (2)−(n^2 /2) ξ_(n−1) (2) +(1/(2(n+1)))  =(n−(1/2))ξ(2)−((2n^2 −2n +1)/2) ξ_(n−1) (2)−(n^2 /2) ξ_(n−1) (2)+(1/(2(n+1)))
$${we}\:{have}\:\xi\left(\mathrm{2}\right)\:=\sum_{{k}=\mathrm{1}} ^{\infty} \:\frac{\mathrm{1}}{{k}^{\mathrm{2}} }\:=\sum_{{k}=\mathrm{1}} ^{{n}} \:\frac{\mathrm{1}}{{k}^{\mathrm{2}} }\:+\sum_{{k}={n}+\mathrm{1}} ^{\infty} \:\frac{\mathrm{1}}{{k}^{\mathrm{2}} }\:=\xi_{{n}} \left(\mathrm{2}\right)\:+\sum_{{k}={n}+\mathrm{1}} ^{\infty} \:\frac{\mathrm{1}}{{k}^{\mathrm{2}} }\:\Rightarrow \\ $$$$\sum_{{k}={n}+\mathrm{1}} ^{\infty} \:\frac{\mathrm{1}}{{k}^{\mathrm{2}} }\:=\xi\left(\mathrm{2}\right)−\xi_{{n}} \left(\mathrm{2}\right)\:\:{also}\:\:\sum_{{k}={n}} ^{\infty} \:\frac{\mathrm{1}}{{k}^{\mathrm{2}} }\:=\xi\left(\mathrm{2}\right)−\xi_{{n}−\mathrm{1}} \left(\mathrm{2}\right)\:\Rightarrow \\ $$$${U}_{{n}} =\frac{{n}^{\mathrm{2}} }{\mathrm{2}}\left(\:\xi\left(\mathrm{2}\right)−\xi_{{n}} \left(\mathrm{2}\right)\right)−\frac{\left({n}−\mathrm{1}\right)^{\mathrm{2}} }{\mathrm{2}}\:\left(\xi\left(\mathrm{2}\right)−\xi_{{n}−\mathrm{1}} \left(\mathrm{2}\right)\right)\:+\frac{{n}^{\mathrm{2}} }{\mathrm{2}\left({n}+\mathrm{1}\right)}\:−\frac{{n}−\mathrm{1}}{\mathrm{2}} \\ $$$$=\frac{{n}^{\mathrm{2}} −\left({n}−\mathrm{1}\right)^{\mathrm{2}} }{\mathrm{2}}\xi\left(\mathrm{2}\right)\:−\frac{{n}^{\mathrm{2}} }{\mathrm{2}}\xi_{{n}} \left(\mathrm{2}\right)−\frac{\left({n}−\mathrm{1}\right)^{\mathrm{2}} }{\mathrm{2}}\:\xi_{{n}−\mathrm{1}} \left(\mathrm{2}\right)\:\:+\frac{{n}^{\mathrm{2}} −\left({n}−\mathrm{1}\right)\left({n}+\mathrm{1}\right)}{\mathrm{2}\left({n}+\mathrm{1}\right)} \\ $$$$=\frac{\mathrm{2}{n}−\mathrm{1}}{\mathrm{2}}\:\xi\left(\mathrm{2}\right)\:−\frac{{n}^{\mathrm{2}} }{\mathrm{2}}\left(\:\xi_{{n}−\mathrm{1}} \left(\mathrm{2}\right)\:+\frac{\mathrm{1}}{{n}^{\mathrm{2}} }\right)−\frac{\left({n}−\mathrm{1}\right)^{\mathrm{2}} }{\mathrm{2}}\:\xi_{{n}−\mathrm{1}} \left(\mathrm{2}\right)\:\:+\:\frac{\mathrm{1}}{\mathrm{2}\left({n}+\mathrm{1}\right)} \\ $$$$=\left({n}−\frac{\mathrm{1}}{\mathrm{2}}\right)\xi\left(\mathrm{2}\right)\:−\frac{{n}^{\mathrm{2}} \:+{n}^{\mathrm{2}} −\mathrm{2}{n}+\mathrm{1}}{\mathrm{2}}\:\xi_{{n}−\mathrm{1}} \left(\mathrm{2}\right)−\frac{{n}^{\mathrm{2}} }{\mathrm{2}}\:\xi_{{n}−\mathrm{1}} \left(\mathrm{2}\right)\:+\frac{\mathrm{1}}{\mathrm{2}\left({n}+\mathrm{1}\right)} \\ $$$$=\left({n}−\frac{\mathrm{1}}{\mathrm{2}}\right)\xi\left(\mathrm{2}\right)−\frac{\mathrm{2}{n}^{\mathrm{2}} −\mathrm{2}{n}\:+\mathrm{1}}{\mathrm{2}}\:\xi_{{n}−\mathrm{1}} \left(\mathrm{2}\right)−\frac{{n}^{\mathrm{2}} }{\mathrm{2}}\:\xi_{{n}−\mathrm{1}} \left(\mathrm{2}\right)+\frac{\mathrm{1}}{\mathrm{2}\left({n}+\mathrm{1}\right)} \\ $$
Commented by maxmathsup by imad last updated on 02/Jun/19
ξ_n (x) =Σ_(k=1) ^n  (1/k^x )    with x>1
$$\xi_{{n}} \left({x}\right)\:=\sum_{{k}=\mathrm{1}} ^{{n}} \:\frac{\mathrm{1}}{{k}^{{x}} }\:\:\:\:{with}\:{x}>\mathrm{1}\: \\ $$

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