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Question Number 133829 by metamorfose last updated on 24/Feb/21

find Σ_(n=1) ^∞ ((x^n sin(nx))/n)=...?

find∑∞n=1xnsin(nx)n=...?

Answered by Dwaipayan Shikari last updated on 24/Feb/21

log(1−xe^(ix) )=log((√((1−xcosx)^2 +x^2 sin^2 x)))+itan^(−1) ((xsinx)/(xcosx−1)) −i(xsin(x)+(x^2 /2)sin(2x)+(x^3 /3)sin(3x)+...)=itan^(−1) ((xsinx)/(xcosx−1)) Σ_(n=1) ^∞ ((x^n sin(nx))/n)=−tan^(−1) ((xsinx)/(xcosx−1))

log(1−xeix)=log((1−xcosx)2+x2sin2x)+itan−1xsinxxcosx−1−i(xsin(x)+x22sin(2x)+x33sin(3x)+...)=itan−1xsinxxcosx−1∑∞n=1xnsin(nx)n=−tan−1xsinxxcosx−1

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