Menu Close

let-f-x-be-f-x-x-ln-1-x-prove-there-exists-a-sequence-a-k-such-that-D-f-x-0-a-k-x-k-f-x-2-




Question Number 161567 by HongKing last updated on 19/Dec/21
let  f(x)  be  f(x) = (x/(ln(1 - x)))  prove there exists a sequence {a_k } such that  D[f(x)] = [Σ_0 ^( ∞)  a_k  x^k ] [f(x)]^2
letf(x)bef(x)=xln(1x)provethereexistsasequence{ak}suchthatD[f(x)]=[0akxk][f(x)]2
Answered by mindispower last updated on 19/Dec/21
what do you mean By D[f(x)]?
whatdoyoumeanByD[f(x)]?
Commented by HongKing last updated on 19/Dec/21
(d/dx)  my dear Sir
ddxmydearSir
Answered by mindispower last updated on 19/Dec/21
Df(x)=((ln(1−x)+(x/(1−x)))/(ln^2 (1−x)))  =(x^2 /(ln^2 (1−x)))(((ln(1−x))/x^2 )+(1/(x(1−x))))  ((ln(1−x))/x^2 )+(1/(x(1−x)))=−(1/x^2 )Σ_(k≥1) (x^k /k)+(1/x)Σ_(k≥1) x^(k−1)   −(1/x)Σ_(k≥1) (x^(k−1) /k)+((Σ_(k≥1) x^(k−1) )/x)  =(1/x)(Σ_(k≥2) x^(k−1) −Σ_(k≥2) (x^(k−1) /k))=Σ_(k≥2) (((k−1)/k)x^(k−2) )  =Σ_(k≥0) ((k+1)/(k+2))x^k   Df(x)=(Σ_(k≥0) ((k+1)/(k+2))x^k )f^2 (x)
Df(x)=ln(1x)+x1xln2(1x)=x2ln2(1x)(ln(1x)x2+1x(1x))ln(1x)x2+1x(1x)=1x2k1xkk+1xk1xk11xk1xk1k+k1xk1x=1x(k2xk1k2xk1k)=k2(k1kxk2)=k0k+1k+2xkDf(x)=(k0k+1k+2xk)f2(x)
Commented by HongKing last updated on 19/Dec/21
A nice solution my dear Sir thank you
AnicesolutionmydearSirthankyou
Commented by HongKing last updated on 20/Dec/21
My dear Sir, can you imagine, what  could we create by writing the derivative  of a function in the way I proposed?
MydearSir,canyouimagine,whatcouldwecreatebywritingthederivativeofafunctioninthewayIproposed?

Leave a Reply

Your email address will not be published. Required fields are marked *