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Question Number 43938 by abdo.msup.com last updated on 18/Sep/18

calvulste A_n =∫_0 ^n  t^2 [(1/((t+1)^3 ))]dt  and lim_(n→+∞)  A_n

$${calvulste}\:{A}_{{n}} =\int_{\mathrm{0}} ^{{n}} \:{t}^{\mathrm{2}} \left[\frac{\mathrm{1}}{\left({t}+\mathrm{1}\right)^{\mathrm{3}} }\right]{dt} \\ $$$${and}\:{lim}_{{n}\rightarrow+\infty} \:{A}_{{n}} \\ $$

Commented by maxmathsup by imad last updated on 20/Sep/18

changement (1/((t+1)^3 )) =x give (t+1)^3 =(1/x) ⇒t+1 = x^(−(1/3))  ⇒  A_n  = ∫_0 ^n (x^(−(1/3)) −1)^2 [x] (−(1/3)) x^(−(4/3)) dx  =−(1/3) ∫_0 ^n  (x^(−(2/3))  −2x^(−(1/3))  +1)x^(−(4/3)) [x] dx  =−(1/3) ∫_0 ^n  {x^(−2)  −2x^(−(5/3))  +x^(−(4/3)) }[x]dx ⇒  −3A_n =Σ_(k=0) ^(n−1)   ∫_k ^(k+1)   (k x^(−2) dx −2k x^(−(5/3))  +k x^(−(4/3)) )dx  =Σ_(k=0) ^(n−1)  k  ∫_k ^(k+1)  x^(−2) dx −2 Σ_(k=0) ^(n−1)  k ∫_k ^(k+1)  x^(−(5/3)) dx  +Σ_(k=0) ^(n−1)  k ∫_k ^(k+1)  x^(−(4/3))  dx  =Σ_(k=0) ^(n−1)  k [−(1/x)]_k ^(k+1) −2 Σ_(k=0) ^(n−1)  k [(1/(−(5/3)+1))x^(−(5/3)+1) ]_k ^(k+1)   +Σ_(k=0) ^(n−1)  k [ (1/(−(4/3)+1)) x^(−(4/3)+1) ]_k ^(k+1)   =Σ_(k=0) ^(n−1)  k{ (1/k)−(1/(k+1))} +3 Σ_(k=0) ^(n−1) k{ (k+1)^(−(2/3))  −k^(−(2/3)) }     −3Σ_(k=0) ^(n−1)  k { (k+1)^(−(1/3))  −k^(−(1/3)) }  = Σ_(k=0) ^(n−1)    (1/(k+1)) +3 { Σ_(k=0) ^(n−1)    (k/((k+1)^(2/3) )) −Σ_(k=0) ^(n−1)     (k/k^(2/3) )}  −3 { Σ_(k=0) ^(n−1)    (k/((k+1)^(1/3) )) −Σ_(k=0) ^(n−1)    (k/k^(1/3) )} ⇒  A_n =Σ_(k=1) ^n  (1/k)  +3 { Σ_(k=0) ^(n−1)    (k/((k+1)^(2/3) )) −Σ_(k=0) ^(n−1)  k^(1/3) }  −3 { Σ_(k=0) ^(n−1)     (k/((k+1)^(1/3) )) −Σ_(k=0) ^(n−1)   k^(2/3)  } .

$${changement}\:\frac{\mathrm{1}}{\left({t}+\mathrm{1}\right)^{\mathrm{3}} }\:={x}\:{give}\:\left({t}+\mathrm{1}\right)^{\mathrm{3}} =\frac{\mathrm{1}}{{x}}\:\Rightarrow{t}+\mathrm{1}\:=\:{x}^{−\frac{\mathrm{1}}{\mathrm{3}}} \:\Rightarrow \\ $$$${A}_{{n}} \:=\:\int_{\mathrm{0}} ^{{n}} \left({x}^{−\frac{\mathrm{1}}{\mathrm{3}}} −\mathrm{1}\right)^{\mathrm{2}} \left[{x}\right]\:\left(−\frac{\mathrm{1}}{\mathrm{3}}\right)\:{x}^{−\frac{\mathrm{4}}{\mathrm{3}}} {dx} \\ $$$$=−\frac{\mathrm{1}}{\mathrm{3}}\:\int_{\mathrm{0}} ^{{n}} \:\left({x}^{−\frac{\mathrm{2}}{\mathrm{3}}} \:−\mathrm{2}{x}^{−\frac{\mathrm{1}}{\mathrm{3}}} \:+\mathrm{1}\right){x}^{−\frac{\mathrm{4}}{\mathrm{3}}} \left[{x}\right]\:{dx} \\ $$$$=−\frac{\mathrm{1}}{\mathrm{3}}\:\int_{\mathrm{0}} ^{{n}} \:\left\{{x}^{−\mathrm{2}} \:−\mathrm{2}{x}^{−\frac{\mathrm{5}}{\mathrm{3}}} \:+{x}^{−\frac{\mathrm{4}}{\mathrm{3}}} \right\}\left[{x}\right]{dx}\:\Rightarrow \\ $$$$−\mathrm{3}{A}_{{n}} =\sum_{{k}=\mathrm{0}} ^{{n}−\mathrm{1}} \:\:\int_{{k}} ^{{k}+\mathrm{1}} \:\:\left({k}\:{x}^{−\mathrm{2}} {dx}\:−\mathrm{2}{k}\:{x}^{−\frac{\mathrm{5}}{\mathrm{3}}} \:+{k}\:{x}^{−\frac{\mathrm{4}}{\mathrm{3}}} \right){dx} \\ $$$$=\sum_{{k}=\mathrm{0}} ^{{n}−\mathrm{1}} \:{k}\:\:\int_{{k}} ^{{k}+\mathrm{1}} \:{x}^{−\mathrm{2}} {dx}\:−\mathrm{2}\:\sum_{{k}=\mathrm{0}} ^{{n}−\mathrm{1}} \:{k}\:\int_{{k}} ^{{k}+\mathrm{1}} \:{x}^{−\frac{\mathrm{5}}{\mathrm{3}}} {dx}\:\:+\sum_{{k}=\mathrm{0}} ^{{n}−\mathrm{1}} \:{k}\:\int_{{k}} ^{{k}+\mathrm{1}} \:{x}^{−\frac{\mathrm{4}}{\mathrm{3}}} \:{dx} \\ $$$$=\sum_{{k}=\mathrm{0}} ^{{n}−\mathrm{1}} \:{k}\:\left[−\frac{\mathrm{1}}{{x}}\right]_{{k}} ^{{k}+\mathrm{1}} −\mathrm{2}\:\sum_{{k}=\mathrm{0}} ^{{n}−\mathrm{1}} \:{k}\:\left[\frac{\mathrm{1}}{−\frac{\mathrm{5}}{\mathrm{3}}+\mathrm{1}}{x}^{−\frac{\mathrm{5}}{\mathrm{3}}+\mathrm{1}} \right]_{{k}} ^{{k}+\mathrm{1}} \\ $$$$+\sum_{{k}=\mathrm{0}} ^{{n}−\mathrm{1}} \:{k}\:\left[\:\frac{\mathrm{1}}{−\frac{\mathrm{4}}{\mathrm{3}}+\mathrm{1}}\:{x}^{−\frac{\mathrm{4}}{\mathrm{3}}+\mathrm{1}} \right]_{{k}} ^{{k}+\mathrm{1}} \\ $$$$=\sum_{{k}=\mathrm{0}} ^{{n}−\mathrm{1}} \:{k}\left\{\:\frac{\mathrm{1}}{{k}}−\frac{\mathrm{1}}{{k}+\mathrm{1}}\right\}\:+\mathrm{3}\:\sum_{{k}=\mathrm{0}} ^{{n}−\mathrm{1}} {k}\left\{\:\left({k}+\mathrm{1}\right)^{−\frac{\mathrm{2}}{\mathrm{3}}} \:−{k}^{−\frac{\mathrm{2}}{\mathrm{3}}} \right\}\:\:\: \\ $$$$−\mathrm{3}\sum_{{k}=\mathrm{0}} ^{{n}−\mathrm{1}} \:{k}\:\left\{\:\left({k}+\mathrm{1}\right)^{−\frac{\mathrm{1}}{\mathrm{3}}} \:−{k}^{−\frac{\mathrm{1}}{\mathrm{3}}} \right\} \\ $$$$=\:\sum_{{k}=\mathrm{0}} ^{{n}−\mathrm{1}} \:\:\:\frac{\mathrm{1}}{{k}+\mathrm{1}}\:+\mathrm{3}\:\left\{\:\sum_{{k}=\mathrm{0}} ^{{n}−\mathrm{1}} \:\:\:\frac{{k}}{\left({k}+\mathrm{1}\right)^{\frac{\mathrm{2}}{\mathrm{3}}} }\:−\sum_{{k}=\mathrm{0}} ^{{n}−\mathrm{1}} \:\:\:\:\frac{{k}}{{k}^{\frac{\mathrm{2}}{\mathrm{3}}} }\right\} \\ $$$$−\mathrm{3}\:\left\{\:\sum_{{k}=\mathrm{0}} ^{{n}−\mathrm{1}} \:\:\:\frac{{k}}{\left({k}+\mathrm{1}\right)^{\frac{\mathrm{1}}{\mathrm{3}}} }\:−\sum_{{k}=\mathrm{0}} ^{{n}−\mathrm{1}} \:\:\:\frac{{k}}{{k}^{\frac{\mathrm{1}}{\mathrm{3}}} }\right\}\:\Rightarrow \\ $$$${A}_{{n}} =\sum_{{k}=\mathrm{1}} ^{{n}} \:\frac{\mathrm{1}}{{k}}\:\:+\mathrm{3}\:\left\{\:\sum_{{k}=\mathrm{0}} ^{{n}−\mathrm{1}} \:\:\:\frac{{k}}{\left({k}+\mathrm{1}\right)^{\frac{\mathrm{2}}{\mathrm{3}}} }\:−\sum_{{k}=\mathrm{0}} ^{{n}−\mathrm{1}} \:{k}^{\frac{\mathrm{1}}{\mathrm{3}}} \right\} \\ $$$$−\mathrm{3}\:\left\{\:\sum_{{k}=\mathrm{0}} ^{{n}−\mathrm{1}} \:\:\:\:\frac{{k}}{\left({k}+\mathrm{1}\right)^{\frac{\mathrm{1}}{\mathrm{3}}} }\:−\sum_{{k}=\mathrm{0}} ^{{n}−\mathrm{1}} \:\:{k}^{\frac{\mathrm{2}}{\mathrm{3}}} \:\right\}\:. \\ $$

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