Question Number 66875 by Cmr 237 last updated on 20/Aug/19
Commented by mathmax by abdo last updated on 20/Aug/19
$${convergence}\:{of}\:{this}\:{serie}\:\:\:{let}\:\varphi\left({t}\right)\:=\frac{\mathrm{1}}{{t}^{\mathrm{3}} {sin}^{\mathrm{2}} {t}}\:\:{with}\:{t}>\mathrm{1} \\ $$$${we}\:{have}\:\varphi^{'} \left({t}\right)\:=\:−\frac{\mathrm{3}{t}^{\mathrm{2}} {sin}^{\mathrm{2}} {t}\:+\mathrm{2}{t}^{\mathrm{3}} {sint}\:{cost}}{{t}^{\mathrm{6}} {sin}^{\mathrm{4}} {t}}\:\:\:\exists{A}>\mathrm{0}\:/\:{for}\:{t}>{A} \\ $$$$\left.\varphi^{'} \left({t}\right)<\mathrm{0}\:\Rightarrow\varphi\:{is}\:{decreazing}\:{on}\:\right]{A},+\infty\left[\:\:{so}\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\frac{\mathrm{1}}{{n}^{\mathrm{3}} {sin}^{\mathrm{2}} {n}}\right. \\ $$$${and}\:\:\int_{\mathrm{1}} ^{+\infty} \:\frac{{dt}}{{t}^{\mathrm{3}} {sin}^{\mathrm{2}} {t}}\:\:{have}\:{the}\:{same}\:{nature}\: \\ $$$$\int_{\mathrm{1}} ^{+\infty} \:\frac{{dt}}{{t}^{\mathrm{3}} {sin}^{\mathrm{2}} {t}}\:\:\:\:{the}\:{function}\:{is}\:{continue}\:{on}\:\left[\mathrm{1},{A}\left[\:{so}\:{integrable}\:{on}\right.\right. \\ $$$$\left[\mathrm{1},{A}\left[\:\:\:{and}\:{at}\:{V}\left(+\infty\right)\:\:{lim}_{{t}\rightarrow+\infty} \:\:\:{t}^{\mathrm{2}} \:×\frac{\mathrm{1}}{{t}^{\mathrm{3}} {sin}^{\mathrm{2}} {t}}\:={lim}_{{t}\rightarrow+\infty} \:\:\frac{\mathrm{1}}{{tsin}^{\mathrm{2}} {t}}\:=\right.\right. \\ $$$$\Rightarrow{the}\:{integral}\:{converges}\:\Rightarrow\:{this}\:{serie}\:{converges}…. \\ $$
Commented by Prithwish sen last updated on 20/Aug/19
$$\frac{\mathrm{1}}{\mathrm{t}^{\mathrm{3}} \mathrm{lim}_{\mathrm{t}\rightarrow\mathrm{0}} \left(\frac{\mathrm{sint}}{\mathrm{t}}\right)^{\mathrm{2}} }\:=\:\frac{\mathrm{1}}{\mathrm{0}.\mathrm{1}}\:\rightarrow\infty \\ $$
Commented by mathmax by abdo last updated on 20/Aug/19
$${no}\:{sir}\:{i}\:{want}\:{to}\:{say}\:{that}\:{lim}_{{t}\rightarrow+\infty} \:\frac{\mathrm{1}}{{t}\:{sin}^{\mathrm{2}} {t}}\:=\mathrm{0} \\ $$
Commented by Prithwish sen last updated on 21/Aug/19
$$\mathrm{oh} \\ $$