Question Number 134439 by Dwaipayan Shikari last updated on 03/Mar/21
![sin1β((sin2)/2)+((sin3)/3)β((sin4)/4)+...=(1/2)](https://www.tinkutara.com/question/Q134439.png)
$${sin}\mathrm{1}β\frac{{sin}\mathrm{2}}{\mathrm{2}}+\frac{{sin}\mathrm{3}}{\mathrm{3}}β\frac{{sin}\mathrm{4}}{\mathrm{4}}+…=\frac{\mathrm{1}}{\mathrm{2}} \\ $$
Answered by mnjuly1970 last updated on 03/Mar/21
![π=Ξ£_(n=1) ^β (((β1)^(nβ1) sin(nx))/n)= =imΞ£_(n=1) ^β (((β1)^(nβ1) (e^(ix) )^n )/n)=im(ln(1+e^(ix) )) =_(z=x+iy) ^(ln(z)) ln((β(x^2 +y^2 )))+itan^(β1) ((y/x)) =im(ln(β((1+cos(x))^2 +sin^2 (x) )) +itan^(β1) (((sin(x))/(1+cos(x)))=tan((x/2))) =(x/2) x:=1 Ξ£_(n=1) ^β (β1)^(nβ1) ((sin(n))/n)=(1/2) ...ββ](https://www.tinkutara.com/question/Q134440.png)
$$\:\:\:\boldsymbol{\Omega}=\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\left(β\mathrm{1}\right)^{{n}β\mathrm{1}} {sin}\left({nx}\right)}{{n}}=\:\:\: \\ $$$$={im}\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\left(β\mathrm{1}\right)^{{n}β\mathrm{1}} \left({e}^{{ix}} \right)^{{n}} }{{n}}={im}\left({ln}\left(\mathrm{1}+{e}^{{ix}} \right)\right) \\ $$$$\:\:\:\underset{{z}={x}+{iy}} {\overset{{ln}\left({z}\right)} {=}}\:{ln}\left(\sqrt{{x}^{\mathrm{2}} +{y}^{\mathrm{2}} }\right)+{itan}^{β\mathrm{1}} \left(\frac{{y}}{{x}}\right) \\ $$$$\:\:\:\:={im}\left({ln}\sqrt{\left(\mathrm{1}+{cos}\left({x}\right)\right)^{\mathrm{2}} +{sin}^{\mathrm{2}} \left({x}\right)\:}\:+{itan}^{β\mathrm{1}} \left(\frac{{sin}\left({x}\right)}{\mathrm{1}+{cos}\left({x}\right)}={tan}\left(\frac{{x}}{\mathrm{2}}\right)\right)\right. \\ $$$$=\frac{{x}}{\mathrm{2}} \\ $$$${x}:=\mathrm{1} \\ $$$$\:\:\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\left(β\mathrm{1}\right)^{{n}β\mathrm{1}} \frac{{sin}\left({n}\right)}{{n}}=\frac{\mathrm{1}}{\mathrm{2}}\:…\checkmark\checkmark \\ $$
Commented by Dwaipayan Shikari last updated on 03/Mar/21
![Great sir! It is simple and nice. (x/2)=sinxβ((sin2x)/2)+((sin3x)/3)β((sin4x)/4)+... (0<x<Ο) But ,Differentiating respect to x (1/2)=cosxβcos2x+cos3xβcos4x+.... (Independent of x!) Is it true? If we take x=0 (1/2)=1β1+1β1+1β1+.... But 0<x<Ο](https://www.tinkutara.com/question/Q134441.png)
$${Great}\:{sir}!\:{It}\:{is}\:{simple}\:{and}\:{nice}. \\ $$$$\frac{{x}}{\mathrm{2}}={sinx}β\frac{{sin}\mathrm{2}{x}}{\mathrm{2}}+\frac{{sin}\mathrm{3}{x}}{\mathrm{3}}β\frac{{sin}\mathrm{4}{x}}{\mathrm{4}}+…\:\:\left(\mathrm{0}<{x}<\pi\right) \\ $$$${But}\:,{Differentiating}\:{respect}\:{to}\:{x} \\ $$$$\frac{\mathrm{1}}{\mathrm{2}}={cosx}β{cos}\mathrm{2}{x}+{cos}\mathrm{3}{x}β{cos}\mathrm{4}{x}+….\:\:\:\left({Independent}\:{of}\:{x}!\right) \\ $$$${Is}\:{it}\:{true}? \\ $$$${If}\:{we}\:{take}\:{x}=\mathrm{0} \\ $$$$\frac{\mathrm{1}}{\mathrm{2}}=\mathrm{1}β\mathrm{1}+\mathrm{1}β\mathrm{1}+\mathrm{1}β\mathrm{1}+….\: \\ $$$${But}\:\mathrm{0}<{x}<\pi \\ $$
Commented by mnjuly1970 last updated on 03/Mar/21
![peace be upon you your effort really really admirable i donβ²t know how old are you by the way your work is very excellent ..with the best wishes allah keep you....](https://www.tinkutara.com/question/Q134443.png)
$${peace}\:{be}\:{upon}\:{you}\: \\ $$$$\:\:{your}\:{effort}\:{really}\:{really}\:{admirable}\: \\ $$$$\:\:{i}\:{don}'{t}\:{know}\:{how}\:{old}\:{are}\:{you} \\ $$$${by}\:{the}\:{way}\:{your}\:{work}\:{is}\:{very} \\ $$$$\:\:{excellent}\:..{with}\:{the}\:{best}\:{wishes} \\ $$$$\:\:\:\:\:\:{allah}\:{keep}\:{you}…. \\ $$