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

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

$${find}\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{{x}^{{n}} {sin}\left({nx}\right)}{{n}}=...? \\ $$

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))

$${log}\left(\mathrm{1}−{xe}^{{ix}} \right)={log}\left(\sqrt{\left(\mathrm{1}−{xcosx}\right)^{\mathrm{2}} +{x}^{\mathrm{2}} {sin}^{\mathrm{2}} {x}}\right)+{itan}^{−\mathrm{1}} \frac{{xsinx}}{{xcosx}−\mathrm{1}} \\ $$$$−{i}\left({xsin}\left({x}\right)+\frac{{x}^{\mathrm{2}} }{\mathrm{2}}{sin}\left(\mathrm{2}{x}\right)+\frac{{x}^{\mathrm{3}} }{\mathrm{3}}{sin}\left(\mathrm{3}{x}\right)+...\right)={itan}^{−\mathrm{1}} \frac{{xsinx}}{{xcosx}−\mathrm{1}} \\ $$$$\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{{x}^{{n}} {sin}\left({nx}\right)}{{n}}=−{tan}^{−\mathrm{1}} \frac{{xsinx}}{{xcosx}−\mathrm{1}} \\ $$

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