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Question Number 74014 by mathmax by abdo last updated on 17/Nov/19
let W(x)=Σ_(1≤i<j≤n) (x^(i+j) /(ij)) calculate W^′ (x).
$$\:\: \\ $$$$\:\:{let}\:{W}\left({x}\right)=\sum_{\mathrm{1}\leqslant{i}<{j}\leqslant{n}} \:\:\frac{{x}^{{i}+{j}} }{{ij}} \\ $$$${calculate}\:{W}\:^{'} \left({x}\right). \\ $$
Commented by mathmax by abdo last updated on 19/Nov/19
we have (Σ_(i=1) ^n (x^i /i))^2 =Σ_(i=1) ^n (x^(2i) /i^2 ) +Σ_(1≤i<j≤n) (x^(i+j) /(ij)) ⇒ W(x)=(Σ_(i=1) ^n (x^i /i))^2 −Σ_(i=1) ^n (x^(2i) /i^2 ) ⇒ W^′ (x)=2(Σ_(i=1) ^n (x^i /i))^′ (Σ_(i=1) ^n (x^i /i))−Σ_(i=1) ^n ((2i x^(2i−1) )/i^2 ) =2(Σ_(i=1) ^n x^(i−1) )(Σ_(i=1) ^n (x^i /i))−2 Σ_(i=1) ^n (x^(2i−1) /i) we have Σ_(i=1) ^n x^(i−1) =Σ_(i=0) ^(n−1) x^i =((1−x^n )/(1−x)) if x≠1 and Σ_(i=1) ^n x^(i−1) =n if x=1 let f(x)=Σ_(i=1) ^n (x^i /i) ⇒f^′ (x)=Σ_(i=1) ^n x^(i−1) =((1−x^n )/(1−x)) (if x≠1) f(x)=∫_0 ^x ((1−t^n )/(1−t)) dt +c (c=f(0)=0) ⇒f(x)=∫_0 ^x ((1−t^n )/(1−t))dt =∫_0 ^x (dt/(1−t)) −∫_0 ^x (t^n /(1−t))dt =−ln∣1−x∣ −∫_0 ^x (t^n /(1−t))dt x=1 ⇒f(x)=Σ_(i=1) ^n (1/i) =H_n also Σ_(i=1) ^n (x^(2i−1) /i) =(1/x)Σ_(i=1) ^n (((x^2 )^i )/i) =(1/x)f(x^2 )=(1/x)(−ln∣1−x^2 ∣−∫_0 ^x^2 (t^n /(1−t))dt) =−(1/x)(ln∣1−x^2 ∣+∫_0 ^x^2 (t^n /(1−t))dt) (x≠1) so for x≠1 W^′ (x)=2(((x^n −1)/(x−1)))(ln∣1−x∣+∫_0 ^x (t^n /(1−t))dt)+(2/x)(ln∣1−x^2 ∣+∫_0 ^x^2 (t^n /(1−t))dt)
$${we}\:{have}\:\left(\sum_{{i}=\mathrm{1}} ^{{n}} \:\frac{{x}^{{i}} }{{i}}\right)^{\mathrm{2}} \:=\sum_{{i}=\mathrm{1}} ^{{n}} \:\frac{{x}^{\mathrm{2}{i}} }{{i}^{\mathrm{2}} }\:+\sum_{\mathrm{1}\leqslant{i}<{j}\leqslant{n}} \:\:\frac{{x}^{{i}+{j}} }{{ij}}\:\Rightarrow \\ $$$${W}\left({x}\right)=\left(\sum_{{i}=\mathrm{1}} ^{{n}} \:\frac{{x}^{{i}} }{{i}}\right)^{\mathrm{2}} −\sum_{{i}=\mathrm{1}} ^{{n}} \:\frac{{x}^{\mathrm{2}{i}} }{{i}^{\mathrm{2}} }\:\Rightarrow \\ $$$${W}\:^{'} \left({x}\right)=\mathrm{2}\left(\sum_{{i}=\mathrm{1}} ^{{n}} \:\frac{{x}^{{i}} }{{i}}\right)^{'} \left(\sum_{{i}=\mathrm{1}} ^{{n}} \:\frac{{x}^{{i}} }{{i}}\right)−\sum_{{i}=\mathrm{1}} ^{{n}} \frac{\mathrm{2}{i}\:{x}^{\mathrm{2}{i}−\mathrm{1}} }{{i}^{\mathrm{2}} } \\ $$$$=\mathrm{2}\left(\sum_{{i}=\mathrm{1}} ^{{n}} \:{x}^{{i}−\mathrm{1}} \right)\left(\sum_{{i}=\mathrm{1}} ^{{n}} \:\frac{{x}^{{i}} }{{i}}\right)−\mathrm{2}\:\sum_{{i}=\mathrm{1}} ^{{n}} \:\frac{{x}^{\mathrm{2}{i}−\mathrm{1}} }{{i}}\:\:{we}\:{have} \\ $$$$\sum_{{i}=\mathrm{1}} ^{{n}} \:{x}^{{i}−\mathrm{1}} =\sum_{{i}=\mathrm{0}} ^{{n}−\mathrm{1}} \:{x}^{{i}} \:\:=\frac{\mathrm{1}−{x}^{{n}} }{\mathrm{1}−{x}}\:\:{if}\:{x}\neq\mathrm{1}\:\:{and}\:\sum_{{i}=\mathrm{1}} ^{{n}} \:{x}^{{i}−\mathrm{1}} ={n}\:{if}\:{x}=\mathrm{1} \\ $$$${let}\:{f}\left({x}\right)=\sum_{{i}=\mathrm{1}} ^{{n}} \:\frac{{x}^{{i}} }{{i}}\:\Rightarrow{f}^{'} \left({x}\right)=\sum_{{i}=\mathrm{1}} ^{{n}} {x}^{{i}−\mathrm{1}} \:=\frac{\mathrm{1}−{x}^{{n}} }{\mathrm{1}−{x}}\:\:\:\left({if}\:{x}\neq\mathrm{1}\right) \\ $$$${f}\left({x}\right)=\int_{\mathrm{0}} ^{{x}} \:\frac{\mathrm{1}−{t}^{{n}} }{\mathrm{1}−{t}}\:{dt}\:\:+{c}\:\:\:\left({c}={f}\left(\mathrm{0}\right)=\mathrm{0}\right)\:\Rightarrow{f}\left({x}\right)=\int_{\mathrm{0}} ^{{x}} \:\frac{\mathrm{1}−{t}^{{n}} }{\mathrm{1}−{t}}{dt} \\ $$$$=\int_{\mathrm{0}} ^{{x}} \:\frac{{dt}}{\mathrm{1}−{t}}\:−\int_{\mathrm{0}} ^{{x}} \:\frac{{t}^{{n}} }{\mathrm{1}−{t}}{dt}\:=−{ln}\mid\mathrm{1}−{x}\mid\:−\int_{\mathrm{0}} ^{{x}} \frac{{t}^{{n}} }{\mathrm{1}−{t}}{dt} \\ $$$${x}=\mathrm{1}\:\Rightarrow{f}\left({x}\right)=\sum_{{i}=\mathrm{1}} ^{{n}} \:\frac{\mathrm{1}}{{i}}\:={H}_{{n}} \:\:\:{also}\:\: \\ $$$$\sum_{{i}=\mathrm{1}} ^{{n}} \:\:\frac{{x}^{\mathrm{2}{i}−\mathrm{1}} }{{i}}\:=\frac{\mathrm{1}}{{x}}\sum_{{i}=\mathrm{1}} ^{{n}} \:\frac{\left({x}^{\mathrm{2}} \right)^{{i}} }{{i}}\:=\frac{\mathrm{1}}{{x}}{f}\left({x}^{\mathrm{2}} \right)=\frac{\mathrm{1}}{{x}}\left(−{ln}\mid\mathrm{1}−{x}^{\mathrm{2}} \mid−\int_{\mathrm{0}} ^{{x}^{\mathrm{2}} } \frac{{t}^{{n}} }{\mathrm{1}−{t}}{dt}\right) \\ $$$$=−\frac{\mathrm{1}}{{x}}\left({ln}\mid\mathrm{1}−{x}^{\mathrm{2}} \mid+\int_{\mathrm{0}} ^{{x}^{\mathrm{2}} } \:\frac{{t}^{{n}} }{\mathrm{1}−{t}}{dt}\right)\:\:\:\left({x}\neq\mathrm{1}\right)\:{so}\:{for}\:{x}\neq\mathrm{1} \\ $$$${W}^{'} \left({x}\right)=\mathrm{2}\left(\frac{{x}^{{n}} −\mathrm{1}}{{x}−\mathrm{1}}\right)\left({ln}\mid\mathrm{1}−{x}\mid+\int_{\mathrm{0}} ^{{x}} \:\frac{{t}^{{n}} }{\mathrm{1}−{t}}{dt}\right)+\frac{\mathrm{2}}{{x}}\left({ln}\mid\mathrm{1}−{x}^{\mathrm{2}} \mid+\int_{\mathrm{0}} ^{{x}^{\mathrm{2}} } \frac{{t}^{{n}} }{\mathrm{1}−{t}}{dt}\right) \\ $$$$ \\ $$$$ \\ $$

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