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find-n-1-1-n-n-cos-nx-and-n-1-1-n-n-sin-nx-




Question Number 38942 by math khazana by abdo last updated on 01/Jul/18
find  Σ_(n=1) ^∞  (((−1)^n )/n)cos(nx) and Σ_(n=1) ^∞  (((−1)^n )/n)sin(nx)
findn=1(1)nncos(nx)andn=1(1)nnsin(nx)
Commented by prof Abdo imad last updated on 09/Jul/18
let f(x)=Σ_(n=1) ^∞  (((−1)^n )/n) cos(nx)  f(x) =Re( Σ_(n=1) ^∞  (((−1)^n )/n) e^(inx) )=Re(w(x))  w(x)=Σ_(n=1) ^∞   (((−e^(ix) )^n )/n) =−ln(1+e^(ix) )  ( Σ (z^n /n) =−ln(1−z))  but  ln(1+e^(ix) ) =ln( 1 +cosx +isinx)  =ln(2cos^2 ((x/2)) +2i sin((x/2))cos((x/2)))  =ln(2) +ln( cos((x/2))e^((ix)/2) )  =ln(2) +ln(cos((x/2))) +((ix)/2) ⇒  w(x)=−ln(2cos((x/2)))−((ix)/2) ⇒  f(x)=−ln(2cos((x/2))) also we get  Σ_(n=1) ^∞   (((−1)^n )/n)sin(nx)=−(x/2)  .
letf(x)=n=1(1)nncos(nx)f(x)=Re(n=1(1)nneinx)=Re(w(x))w(x)=n=1(eix)nn=ln(1+eix)(Σznn=ln(1z))butln(1+eix)=ln(1+cosx+isinx)=ln(2cos2(x2)+2isin(x2)cos(x2))=ln(2)+ln(cos(x2)eix2)=ln(2)+ln(cos(x2))+ix2w(x)=ln(2cos(x2))ix2f(x)=ln(2cos(x2))alsowegetn=1(1)nnsin(nx)=x2.

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