Question Number 210441 by universe last updated on 09/Aug/24
Answered by aleks041103 last updated on 09/Aug/24
$$\left({i}\right)\: \\ $$$${for}\:{odd}\:{n},\:{the}\:{function}\:{sin}^{{n}} \left({nx}\right)\:{is}\:{similar} \\ $$$${to}\:{sin}\left({x}\right)\:{with}\:{the}\:{fact}\:{that}\:{the}\:{primitive}\:{function}\:{oscilates} \\ $$$${between}\:\mathrm{0}\:{and}\:{the}\:{maximal}\:{possible}\:{value} \\ $$$$\int_{\mathrm{0}} ^{\:\pi/{n}} {sin}^{{n}} \left({nx}\right){dx}\:=\:\frac{\mathrm{1}}{{n}}\int_{\mathrm{0}} ^{\:\pi} {sin}^{{n}} \left({x}\right){dx}\leqslant \\ $$$$\leqslant\frac{\mathrm{1}}{{n}}\int_{\mathrm{0}} ^{\:\pi} {sin}\left({x}\right){dx}=\frac{\mathrm{2}}{{n}} \\ $$$$ \\ $$$${s}_{{n}} ={F}_{{n}} \left(\mathrm{1}\right)−{F}_{{n}} \left(\mathrm{0}\right) \\ $$$${for}\:{odd}\:{n}\:{we}\:{have}\:{F}_{{n}} \left({x}\right)\in\left[\mathrm{0},\mathrm{2}/{n}\right] \\ $$$${then}\:{s}_{{n}} \leqslant\frac{\mathrm{2}}{{n}},\:{odd}\:{n}. \\ $$
Answered by aleks041103 last updated on 09/Aug/24
$$\left({ii}\right)\:{for}\:{even}\:{n}: \\ $$$$\int_{\mathrm{0}} ^{\:\mathrm{1}} {sin}^{{n}} \left({nx}\right){dx}=\frac{\mathrm{1}}{{n}}\int_{\mathrm{0}} ^{\:{n}} {sin}^{{n}} \left({x}\right){dx}= \\ $$$$=\frac{\mathrm{1}}{{n}}\left[\underset{{k}=\mathrm{0}} {\overset{\lfloor{n}/\pi\rfloor−\mathrm{1}} {\sum}}\int_{\pi{k}} ^{\:\pi\left({k}+\mathrm{1}\right)} {sin}^{{n}} \left({x}\right){dx}\:+\:\int_{\pi\lfloor{n}/\pi\rfloor} ^{\:{n}} {sin}^{{n}} \left({x}\right){dx}\right]= \\ $$$$=\frac{\mathrm{1}}{{n}}\left(\lfloor\frac{{n}}{\pi}\rfloor+{c}\right)\int_{\mathrm{0}} ^{\:\pi} {sin}^{{n}} \left({x}\right){dx} \\ $$$${where}\:\mathrm{0}<{c}<\mathrm{1}. \\ $$$$ \\ $$$${now}\:{let} \\ $$$${I}_{{n}} =\int_{\mathrm{0}} ^{\:\pi} {sin}^{\mathrm{2}{n}} \left({x}\right){dx}= \\ $$$$=\int_{\mathrm{0}} ^{\:\pi} {sin}^{\mathrm{2}{n}−\mathrm{2}} \left({x}\right){dx}\:−\:\int_{\mathrm{0}} ^{\:\pi} {cos}\left({x}\right){d}\left(\frac{{sin}^{\mathrm{2}{n}−\mathrm{1}} \left({x}\right)}{\mathrm{2}{n}−\mathrm{1}}\right){dx}= \\ $$$$={I}_{{n}−\mathrm{1}} −\left[\frac{{sin}^{\mathrm{2}{n}−\mathrm{1}} \left({x}\right){cos}\left({x}\right)}{\mathrm{2}{n}−\mathrm{1}}\right]_{\mathrm{0}} ^{\pi} −\frac{\mathrm{1}}{\mathrm{2}{n}−\mathrm{1}}\int_{\mathrm{0}} ^{\:\pi} {sin}^{\mathrm{2}{n}} \left({x}\right){dx}= \\ $$$$={I}_{{n}−\mathrm{1}} −\frac{\mathrm{1}}{\mathrm{2}{n}−\mathrm{1}}{I}_{{n}} \\ $$$$\Rightarrow{I}_{{n}−\mathrm{1}} =\frac{\mathrm{2}{n}}{\mathrm{2}{n}−\mathrm{1}}{I}_{{n}} \\ $$$$\Rightarrow{I}_{{n}} =\frac{\mathrm{2}{n}−\mathrm{1}}{\mathrm{2}{n}}{I}_{{n}−\mathrm{1}} \\ $$$$\Rightarrow{I}_{{n}} =\frac{\left(\mathrm{2}{n}−\mathrm{1}\right)\left(\mathrm{2}{n}−\mathrm{3}\right)…\mathrm{3}.\mathrm{1}}{\left(\mathrm{2}{n}\right)\left(\mathrm{2}{n}−\mathrm{2}\right)…\mathrm{4}.\mathrm{2}}{I}_{\mathrm{0}} \\ $$$${I}_{\mathrm{0}} =\int_{\mathrm{0}} ^{\:\pi} {dx}=\pi \\ $$$$\Rightarrow{I}_{{n}} =\frac{\left(\mathrm{2}{n}−\mathrm{1}\right)!!}{\left(\mathrm{2}{n}\right)!!}\pi \\ $$$$\left(\mathrm{2}{n}\right)!!=\mathrm{2}^{{n}} {n}! \\ $$$$\left(\mathrm{2}{n}−\mathrm{1}\right)!!=\frac{\left(\mathrm{2}{n}\right)!}{\left(\mathrm{2}{n}\right)!!}=\frac{\left(\mathrm{2}{n}\right)!}{\mathrm{2}^{{n}} {n}!} \\ $$$$\Rightarrow{I}_{{n}} =\frac{\left(\mathrm{2}{n}\right)!}{\mathrm{4}^{{n}} \left({n}!\right)^{\mathrm{2}} }\pi \\ $$$$\Rightarrow{s}_{\mathrm{2}{n}} =\frac{\mathrm{1}}{\mathrm{2}{n}}\left(\lfloor\frac{\mathrm{2}{n}}{\pi}\rfloor+{c}_{{n}} \right){I}_{{n}} =\frac{\pi}{\mathrm{2}{n}}\left(\lfloor\frac{\mathrm{2}{n}}{\pi}\rfloor+{c}_{{n}} \right)\frac{\left(\mathrm{2}{n}\right)!}{\mathrm{4}^{{n}} \left({n}!\right)^{\mathrm{2}} } \\ $$$${Stirling}: \\ $$$${x}!\:\sim\:\sqrt{\mathrm{2}\pi{x}}\:\left(\frac{{x}}{{e}}\right)^{{x}} \\ $$$$\Rightarrow\left(\mathrm{2}{n}\right)!\:\sim\:\sqrt{\mathrm{4}\pi{n}}\left(\frac{\mathrm{2}{n}}{{e}}\right)^{\mathrm{2}{n}} =\sqrt{\mathrm{4}\pi{n}\:}\mathrm{4}^{{n}} \:\left(\frac{{n}}{{e}}\right)^{\mathrm{2}{n}} \\ $$$$\Rightarrow\left({n}!\right)^{\mathrm{2}} \:\sim\:\mathrm{2}\pi{n}\left(\frac{{n}}{{e}}\right)^{\mathrm{2}{n}} \\ $$$$\Rightarrow{s}_{\mathrm{2}{n}} =\frac{\pi}{\mathrm{2}{n}}\left(\lfloor\frac{\mathrm{2}{n}}{\pi}\rfloor+{c}_{{n}} \right)\frac{\left(\mathrm{2}{n}\right)!}{\mathrm{4}^{{n}} \left({n}!\right)^{\mathrm{2}} }\:\sim\:\frac{\pi}{\mathrm{2}{n}}\left(\lfloor\frac{\mathrm{2}{n}}{\pi}\rfloor+{c}_{{n}} \right)\frac{\sqrt{\mathrm{4}\pi{n}}}{\mathrm{2}\pi{n}} \\ $$$$\Rightarrow{s}_{\mathrm{2}{n}} \sim\:\frac{\pi}{\mathrm{2}{n}}\left(\lfloor\frac{\mathrm{2}{n}}{\pi}\rfloor+{c}_{{n}} \right)\frac{\mathrm{1}}{\:\sqrt{\pi{n}}} \\ $$$${it}\:{is}\:{easy}\:{to}\:{see}\:{that}\:\frac{\pi}{\mathrm{2}{n}}\lfloor\frac{\mathrm{2}{n}}{\pi}\rfloor\sim\mathrm{1} \\ $$$$\Rightarrow{s}_{\mathrm{2}{n}} \sim\:\frac{\mathrm{1}}{\:\sqrt{\pi{n}}} \\ $$$$\Rightarrow{s}_{\mathrm{2}{n}} \rightarrow\mathrm{0} \\ $$$$ \\ $$$${from}\:\left({i}\right)\:{we}\:{have}\:\mathrm{0}<{s}_{\mathrm{2}{n}+\mathrm{1}} <\frac{\mathrm{2}}{\mathrm{2}{n}+\mathrm{1}} \\ $$$$\Rightarrow{s}_{\mathrm{2}{n}+\mathrm{1}} \rightarrow\mathrm{0} \\ $$$$ \\ $$$$\Rightarrow{s}_{{n}} \rightarrow\mathrm{0} \\ $$$$\Rightarrow\:{the}\:{only}\:{limiting}\:{point}\:{is}\:\mathrm{0}. \\ $$
Commented by hardmath last updated on 11/Aug/24
$$ \\ $$Dear professor, your solutions are perfect