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Question Number 37271 by abdo.msup.com last updated on 11/Jun/18

find  A_n =∫_1 ^2 ( 1 +(1/x) +(1/x^2 ) +...+(1/x^n ))^2 dx

$${find}\:\:{A}_{{n}} =\int_{\mathrm{1}} ^{\mathrm{2}} \left(\:\mathrm{1}\:+\frac{\mathrm{1}}{{x}}\:+\frac{\mathrm{1}}{{x}^{\mathrm{2}} }\:+...+\frac{\mathrm{1}}{{x}^{{n}} }\right)^{\mathrm{2}} {dx} \\ $$

Commented by prof Abdo imad last updated on 18/Jun/18

let  a_i = (1/x^i )  with 0≤i≤n ⇒  (1+(1/x) +(1/x^2 ) +....+(1/x^n ))^2  =(Σ_(i=0) ^n  a_i )^2   = Σ_(i=0) ^n  a_i ^2    +2 Σ_(0≤i<j≤n) a_i .a_j   =Σ_(i=0) ^n   (1/x^(2i) ) +2 Σ_(0≤i<j≤n)   (1/x^(i+j) ) ⇒  A_n  = ∫_1 ^2  (Σ_(i=0) ^n  (1/x^(2i) ))dx +2 ∫_1 ^2 Σ_(0≤i<j≤n)  (1/x^(i+j) )dx  =Σ_(i=0) ^n  ∫_1 ^2    x^(−2i) dx +2 Σ_(0≤i<j≤n) ∫_1 ^2  x^(−(i+j)) dx  Σ_(i=0) ^n  (1/(−2i+1))[ x^(−2i+1) ]_1 ^2   +2 Σ_(0≤i<j≤n)  (1/(1−(i+j)))[ x^(1−(i+j)) ]_1 ^2   =−Σ_(i=0) ^n   (1/(2i−1)){ 2^(−2i+1)  −1}  −Σ_(0≤i<j≤n)  (1/(i+j−1)){ 2^(1−(i+j))  −1}

$${let}\:\:{a}_{{i}} =\:\frac{\mathrm{1}}{{x}^{{i}} }\:\:{with}\:\mathrm{0}\leqslant{i}\leqslant{n}\:\Rightarrow \\ $$$$\left(\mathrm{1}+\frac{\mathrm{1}}{{x}}\:+\frac{\mathrm{1}}{{x}^{\mathrm{2}} }\:+....+\frac{\mathrm{1}}{{x}^{{n}} }\right)^{\mathrm{2}} \:=\left(\sum_{{i}=\mathrm{0}} ^{{n}} \:{a}_{{i}} \right)^{\mathrm{2}} \\ $$$$=\:\sum_{{i}=\mathrm{0}} ^{{n}} \:{a}_{{i}} ^{\mathrm{2}} \:\:\:+\mathrm{2}\:\sum_{\mathrm{0}\leqslant{i}<{j}\leqslant{n}} {a}_{{i}} .{a}_{{j}} \\ $$$$=\sum_{{i}=\mathrm{0}} ^{{n}} \:\:\frac{\mathrm{1}}{{x}^{\mathrm{2}{i}} }\:+\mathrm{2}\:\sum_{\mathrm{0}\leqslant{i}<{j}\leqslant{n}} \:\:\frac{\mathrm{1}}{{x}^{{i}+{j}} }\:\Rightarrow \\ $$$${A}_{{n}} \:=\:\int_{\mathrm{1}} ^{\mathrm{2}} \:\left(\sum_{{i}=\mathrm{0}} ^{{n}} \:\frac{\mathrm{1}}{{x}^{\mathrm{2}{i}} }\right){dx}\:+\mathrm{2}\:\int_{\mathrm{1}} ^{\mathrm{2}} \sum_{\mathrm{0}\leqslant{i}<{j}\leqslant{n}} \:\frac{\mathrm{1}}{{x}^{{i}+{j}} }{dx} \\ $$$$=\sum_{{i}=\mathrm{0}} ^{{n}} \:\int_{\mathrm{1}} ^{\mathrm{2}} \:\:\:{x}^{−\mathrm{2}{i}} {dx}\:+\mathrm{2}\:\sum_{\mathrm{0}\leqslant{i}<{j}\leqslant{n}} \int_{\mathrm{1}} ^{\mathrm{2}} \:{x}^{−\left({i}+{j}\right)} {dx} \\ $$$$\sum_{{i}=\mathrm{0}} ^{{n}} \:\frac{\mathrm{1}}{−\mathrm{2}{i}+\mathrm{1}}\left[\:{x}^{−\mathrm{2}{i}+\mathrm{1}} \right]_{\mathrm{1}} ^{\mathrm{2}} \\ $$$$+\mathrm{2}\:\sum_{\mathrm{0}\leqslant{i}<{j}\leqslant{n}} \:\frac{\mathrm{1}}{\mathrm{1}−\left({i}+{j}\right)}\left[\:{x}^{\mathrm{1}−\left({i}+{j}\right)} \right]_{\mathrm{1}} ^{\mathrm{2}} \\ $$$$=−\sum_{{i}=\mathrm{0}} ^{{n}} \:\:\frac{\mathrm{1}}{\mathrm{2}{i}−\mathrm{1}}\left\{\:\mathrm{2}^{−\mathrm{2}{i}+\mathrm{1}} \:−\mathrm{1}\right\} \\ $$$$−\sum_{\mathrm{0}\leqslant{i}<{j}\leqslant{n}} \:\frac{\mathrm{1}}{{i}+{j}−\mathrm{1}}\left\{\:\mathrm{2}^{\mathrm{1}−\left({i}+{j}\right)} \:−\mathrm{1}\right\} \\ $$

Commented by math khazana by abdo last updated on 18/Jun/18

A_n = Σ_(i=0) ^n  ((1−2^(−2i+1) )/(2i−1))  +Σ_(0≤i<j≤n) ((1−2^(1−i−j) )/(i+j−1))  .

$${A}_{{n}} =\:\sum_{{i}=\mathrm{0}} ^{{n}} \:\frac{\mathrm{1}−\mathrm{2}^{−\mathrm{2}{i}+\mathrm{1}} }{\mathrm{2}{i}−\mathrm{1}}\:\:+\sum_{\mathrm{0}\leqslant{i}<{j}\leqslant{n}} \frac{\mathrm{1}−\mathrm{2}^{\mathrm{1}−{i}−{j}} }{{i}+{j}−\mathrm{1}}\:\:. \\ $$

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