Question Number 136420 by Dwaipayan Shikari last updated on 21/Mar/21
$$\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{{sin}\left(\mathrm{2}{n}+\mathrm{1}\right)\theta}{\left(\mathrm{2}{n}+\mathrm{1}\right)^{\mathrm{2}} }=\frac{\theta}{\mathrm{2}}{log}\left(\mathrm{2}\right)+{sin}\theta\frac{{log}\left({sin}\theta\right)}{\mathrm{4}}\:_{\mathrm{2}} {F}_{\mathrm{1}} \left(\frac{\mathrm{1}}{\mathrm{2}},\frac{\mathrm{1}}{\mathrm{2}};\frac{\mathrm{3}}{\mathrm{2}};{sin}^{\mathrm{2}} \theta\right)+\frac{{sin}\theta}{\mathrm{16}}\:_{\mathrm{2}} {F}_{\mathrm{1}} \left(\frac{\mathrm{1}}{\mathrm{2}},\frac{\mathrm{1}}{\mathrm{2}},\frac{\mathrm{1}}{\mathrm{2}};\frac{\mathrm{3}}{\mathrm{2}};\frac{\mathrm{3}}{\mathrm{2}};{sin}^{\mathrm{2}} \theta\right) \\ $$$${Prove}\:{or}\:{disprove}\:\:\:\:\:\mathrm{0}<\theta<\pi \\ $$
Answered by mindispower last updated on 21/Mar/21
$$\theta=\pi,{sin}\left(\mathrm{2}{n}+\mathrm{1}\right)\theta=\mathrm{0} \\ $$$${left}=\mathrm{0} \\ $$$${right}=\frac{\pi{ln}\left(\mathrm{2}\right)}{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{4}}+\frac{\mathrm{1}}{\mathrm{16}} \\ $$
Commented by Dwaipayan Shikari last updated on 21/Mar/21
$${Sorry}\:{i}\:{have}\:{edited}\:{the}\:{question} \\ $$
Commented by Dwaipayan Shikari last updated on 21/Mar/21
$${Valid}\:{for}\:\:\:\mathrm{0}<\theta<\pi \\ $$
Commented by mindispower last updated on 22/Mar/21
$${ok}\:{i}\:{will}\:{try}\:{but}\:\:{tchek}\:{it}\:{again} \\ $$
Answered by mindispower last updated on 22/Mar/21
$$\underset{{n}\geqslant\mathrm{0}} {\sum}\frac{{sin}\left(\mathrm{2}{n}+\mathrm{1}\right)\theta}{\left(\mathrm{2}{n}+\mathrm{1}\right)^{\mathrm{2}} }\int={S}=\underset{{n}\geqslant\mathrm{1}} {\sum}\frac{{sin}\left({n}\theta\right)}{{n}^{\mathrm{2}} }−\underset{{n}\geqslant\mathrm{1}} {\sum}\frac{{sin}\left(\mathrm{2}\theta\right)}{\mathrm{4}{n}^{\mathrm{2}} } \\ $$$${recal}\:\underset{{n}\geqslant\mathrm{1}} {\sum}\frac{{sin}\left({n}\theta\right)}{{n}^{\mathrm{2}} }={Cl}_{\mathrm{2}} \left(\theta\right),{clausen}\:{function} \\ $$$${we}\:{get}\:{S}={Cl}_{\mathrm{2}} \left(\theta\right)−\frac{{Cl}_{\mathrm{2}} \left(\mathrm{2}\theta\right)}{\mathrm{4}} \\ $$$${clausen}\:{integration}\:\:{representation} \\ $$$${Cl}_{\mathrm{2}} \left(\theta\right)=−\mathrm{2}\int_{\mathrm{0}} ^{\frac{\theta}{\mathrm{2}}} {ln}\left(\mathrm{2}{sin}\left({t}\right)\right){dt} \\ $$$$={u}={sin}\left({t}\right)=−\mathrm{2}\int_{\mathrm{0}} ^{{sin}\left(\frac{\theta}{\mathrm{2}}\right)} \frac{{ln}\left(\mathrm{2}{u}\right)}{\:\sqrt{\mathrm{1}−{u}^{\mathrm{2}} }}{du}={cl}_{\mathrm{2}} \left(\theta\right) \\ $$$$\left(\mathrm{1}+{x}\right)^{{a}} =\mathrm{1}+\underset{{n}\geqslant\mathrm{1}} {\sum}\frac{\underset{{k}\leqslant{n}−\mathrm{1}} {\prod}\left({a}−{k}\right)}{{n}!}.{x}^{{n}} \\ $$$${cl}_{\mathrm{2}} \left(\theta\right)=−\mathrm{2}\int_{\mathrm{0}} ^{{sin}\left(\frac{\theta}{\mathrm{2}}\right)} \left(\mathrm{1}+\underset{{n}\geqslant\mathrm{1}} {\sum}\left(−\mathrm{1}\right)^{{n}} \frac{\underset{{k}\leqslant{n}−\mathrm{1}} {\prod}\left(−\frac{\mathrm{1}}{\mathrm{2}}−{k}\right){u}^{\mathrm{2}{n}} }{{n}!}\right){ln}\left(\mathrm{2}{u}\right){du} \\ $$$$\int{u}^{\mathrm{2}{n}} {ln}\left(\mathrm{2}{u}\right){du}=\frac{{u}^{\mathrm{2}{n}+\mathrm{1}} }{\mathrm{2}{n}+\mathrm{1}}{ln}\left(\mathrm{2}{u}\right)−\frac{{u}^{\mathrm{2}{n}+\mathrm{1}} }{\left(\mathrm{2}{n}+\mathrm{1}\right)^{\mathrm{2}} } \\ $$$${it}\:{seems}\:{good}\:{starte}\:{we}\:{can}\:{fin}\:_{{p}} {F}_{{q}} \:\: \\ $$$${sinlog}\left({sin}..\right)\:{also}\:{i}\:{think}\:{verry}\:{long}\:{worck} \\ $$$$=−\mathrm{2}\left(\mathrm{1}+\underset{{n}\geqslant\mathrm{1}} {\sum}\frac{\left(\frac{\mathrm{1}}{\mathrm{2}}\right)_{{n}} }{{n}!}\right)\left(\frac{{sin}^{\mathrm{2}{n}+\mathrm{1}} \left(\frac{\theta}{\mathrm{2}}\right){ln}\left(\mathrm{2}{sin}\left(\frac{\theta}{\mathrm{2}}\right)\right)}{\left(\mathrm{2}{n}+\mathrm{1}\right)}−\frac{{sin}^{\mathrm{2}{n}+\mathrm{1}} \left(\frac{\theta}{\mathrm{2}}\right)}{\left(\mathrm{2}{n}+\mathrm{1}\right)^{\mathrm{2}} }\right) \\ $$$$=−\mathrm{2}\left(\mathrm{1}+\underset{{n}\geqslant\mathrm{1}} {\sum}\frac{\left(\frac{\mathrm{1}}{\mathrm{2}}\right)_{{n}} \left(\frac{\mathrm{1}}{\mathrm{2}}\right)_{{n}} }{\mathrm{2}.\left(\frac{\mathrm{3}}{\mathrm{2}}\right)_{{n}} }\right. \\ $$
Commented by Dwaipayan Shikari last updated on 22/Mar/21
$${Thanks}\:{sir}!\:{I}\:{will}\:{post}\:{my}\:{work}\:{later} \\ $$
Commented by mindispower last updated on 22/Mar/21
$${you}\:{are}\:{welcom} \\ $$