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Question Number 134328 by BHOOPENDRA last updated on 02/Mar/21

$$expressf\left(x\right)=xasasineseries$$ $$in0<x<\pi ?$$

Answered by Dwaipayan Shikari last updated on 02/Mar/21

$$log(1+e^{ix})=log\left(\sqrt{2+2sinx}\right)+{itan}^{-1}\frac{sinx}{cosx+1}$$ $$(cosx-\frac{cos2x}{2}+\frac{cos3x}{3}-\frac{cos4x}{4}+...)+i(sinx-\frac{sin2x}{2}+\frac{sin3x}{3}-..)$$ $$=log\left(\sqrt{2+2sinx}\right)+{itan}^{-1}\frac{2sin\frac{x}{2}cos\frac{x}{2}}{2{cos}^2\frac{x}{2}}$$ $$sinx-\frac{sin2x}{2}+\frac{sin3x}{3}-\frac{sin4x}{4}+...={tan}^{-1}\left(tan\frac{x}{2}\right)$$ $$\frac{x}{2}=sinx-\frac{sin2x}{2}+\frac{sin3x}{3}-\frac{sin4x}{4}+...$$

Commented byDwaipayan Shikari last updated on 02/Mar/21

$$Butforx=\pi it{doesn}^{\prime }tholdtrue.0<x<\pi$$ $$forexample$$ $$\frac{\left(\frac{\pi }{2}\right)}{2}=sin\left(\frac{\pi }{2}\right)-\frac{sin\left(\pi \right)}{2}+\frac{sin\left(\frac{3\pi }{2}\right)}{3}-\frac{sin\left(2\pi \right)}{4}+..$$ $$\frac{\pi }{4}=1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+...$$

Commented byBHOOPENDRA last updated on 02/Mar/21

$$thankssir$$

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