Question Number 67381 by mathmax by abdo last updated on 26/Aug/19
$${let}\:{f}\left({x}\right)\:={x}^{\mathrm{2}} \:\:\:\:\:\:\:\mathrm{2}\pi\:{periodic}\:\:{even}\:\:{develop}\:{f}\:{at}\:{fourier}\:{serie} \\ $$
Commented by mathmax by abdo last updated on 27/Aug/19
$${f}\:{even}\:\Rightarrow{f}\left({x}\right)\:=\frac{{a}_{\mathrm{0}} }{\mathrm{2}}\:+\sum_{{n}=\mathrm{1}} ^{\infty} \:{a}_{{n}} {cos}\left({nx}\right)\:\:{with} \\ $$$${a}_{{n}} =\frac{\mathrm{2}}{{T}}\int_{\left[{T}\right]} \:\:{x}^{\mathrm{2}} \:{cos}\left({nx}\right){dx}\:=\frac{\mathrm{2}}{\mathrm{2}\pi}\int_{−\pi} ^{\pi} \:{x}^{\mathrm{2}} {coz}\left({nx}\right){dx} \\ $$$$=\frac{\mathrm{2}}{\pi}\int_{\mathrm{0}} ^{\pi} \:{x}^{\mathrm{2}} {cos}\left({nx}\right){dx}\:\Rightarrow\frac{\pi}{\mathrm{2}}{a}_{{n}} =\int_{\mathrm{0}} ^{\pi} \:{x}^{\mathrm{2}} \:{cos}\left({nx}\right){dx}\:\:{by}\:{parts} \\ $$$$\int_{\mathrm{0}} ^{\pi} \:{x}^{\mathrm{2}} \:{cos}\left({nx}\right){dx}\:=\left[\frac{{x}^{\mathrm{2}} }{{n}}{sin}\left({nx}\right)\right]_{\mathrm{0}} ^{\pi} \:−\int_{\mathrm{0}} ^{\pi} \:\frac{\mathrm{2}{x}}{{n}}{sin}\left({nx}\right){dx} \\ $$$$−\frac{\mathrm{2}}{{n}}\:\int_{\mathrm{0}} ^{\pi} \:{xsin}\left({nx}\right){dx}\:=−\frac{\mathrm{2}}{{n}}\left[\:\:\:\:\left[−\frac{{x}}{{n}}{cos}\left({nx}\right)\right]_{\mathrm{0}} ^{\pi} \:−\int_{\mathrm{0}} ^{\pi} −\frac{\mathrm{1}}{{n}}{cos}\left({nx}\right){dx}\right\} \\ $$$$=−\frac{\mathrm{2}}{{n}}\left\{−\frac{\pi}{{n}}\:\left(−\mathrm{1}\right)^{{n}} \:+\frac{\mathrm{1}}{{n}^{\mathrm{2}} }\left[{sinnx}\right]_{\mathrm{0}} ^{\pi} \right\}\:=\frac{\mathrm{2}\pi}{{n}^{\mathrm{2}} }\left(−\mathrm{1}\right)^{{n}} \:\Rightarrow \\ $$$${a}_{{n}} =\frac{\mathrm{2}}{\pi}×\frac{\mathrm{2}\pi}{{n}^{\mathrm{2}} }\left(−\mathrm{1}\right)^{{n}\:} \:=\frac{\mathrm{4}\pi}{\pi{n}^{\mathrm{2}} }\left(−\mathrm{1}\right)^{{n}} \\ $$$${a}_{\mathrm{0}} =\frac{\mathrm{2}}{\pi}\:\int_{\mathrm{0}} ^{\pi} \:{x}^{\mathrm{2}} \:{dx}\:=\frac{\mathrm{2}}{\pi}\left[\frac{{x}^{\mathrm{3}} }{\mathrm{3}}\right]_{\mathrm{0}} ^{\pi} \:=\frac{\mathrm{2}}{\pi}×\frac{\pi^{\mathrm{3}} }{\mathrm{3}}\:=\frac{\mathrm{2}\pi^{\mathrm{2}} }{\mathrm{3}}\:\Rightarrow \\ $$$$\bigstar{x}^{\mathrm{2}} \:=\frac{\pi^{\mathrm{2}} }{\mathrm{3}}\:+\mathrm{4}\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\frac{\left(−\mathrm{1}\right)^{{n}} }{{n}^{\mathrm{2}} }{cos}\left({nx}\right)\bigstar \\ $$$$ \\ $$