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let-f-x-x-2-2pi-periodic-even-develop-f-at-fourier-serie-




Question Number 67381 by mathmax by abdo last updated on 26/Aug/19
let f(x) =x^2        2π periodic  even  develop f at fourier serie
$${let}\:{f}\left({x}\right)\:={x}^{\mathrm{2}} \:\:\:\:\:\:\:\mathrm{2}\pi\:{periodic}\:\:{even}\:\:{develop}\:{f}\:{at}\:{fourier}\:{serie} \\ $$
Commented by mathmax by abdo last updated on 27/Aug/19
f even ⇒f(x) =(a_0 /2) +Σ_(n=1) ^∞  a_n cos(nx)  with  a_n =(2/T)∫_([T])   x^2  cos(nx)dx =(2/(2π))∫_(−π) ^π  x^2 coz(nx)dx  =(2/π)∫_0 ^π  x^2 cos(nx)dx ⇒(π/2)a_n =∫_0 ^π  x^2  cos(nx)dx  by parts  ∫_0 ^π  x^2  cos(nx)dx =[(x^2 /n)sin(nx)]_0 ^π  −∫_0 ^π  ((2x)/n)sin(nx)dx  −(2/n) ∫_0 ^π  xsin(nx)dx =−(2/n)[    [−(x/n)cos(nx)]_0 ^π  −∫_0 ^π −(1/n)cos(nx)dx}  =−(2/n){−(π/n) (−1)^n  +(1/n^2 )[sinnx]_0 ^π } =((2π)/n^2 )(−1)^n  ⇒  a_n =(2/π)×((2π)/n^2 )(−1)^(n )  =((4π)/(πn^2 ))(−1)^n   a_0 =(2/π) ∫_0 ^π  x^2  dx =(2/π)[(x^3 /3)]_0 ^π  =(2/π)×(π^3 /3) =((2π^2 )/3) ⇒  ★x^2  =(π^2 /3) +4Σ_(n=1) ^∞   (((−1)^n )/n^2 )cos(nx)★
$${f}\:{even}\:\Rightarrow{f}\left({x}\right)\:=\frac{{a}_{\mathrm{0}} }{\mathrm{2}}\:+\sum_{{n}=\mathrm{1}} ^{\infty} \:{a}_{{n}} {cos}\left({nx}\right)\:\:{with} \\ $$$${a}_{{n}} =\frac{\mathrm{2}}{{T}}\int_{\left[{T}\right]} \:\:{x}^{\mathrm{2}} \:{cos}\left({nx}\right){dx}\:=\frac{\mathrm{2}}{\mathrm{2}\pi}\int_{−\pi} ^{\pi} \:{x}^{\mathrm{2}} {coz}\left({nx}\right){dx} \\ $$$$=\frac{\mathrm{2}}{\pi}\int_{\mathrm{0}} ^{\pi} \:{x}^{\mathrm{2}} {cos}\left({nx}\right){dx}\:\Rightarrow\frac{\pi}{\mathrm{2}}{a}_{{n}} =\int_{\mathrm{0}} ^{\pi} \:{x}^{\mathrm{2}} \:{cos}\left({nx}\right){dx}\:\:{by}\:{parts} \\ $$$$\int_{\mathrm{0}} ^{\pi} \:{x}^{\mathrm{2}} \:{cos}\left({nx}\right){dx}\:=\left[\frac{{x}^{\mathrm{2}} }{{n}}{sin}\left({nx}\right)\right]_{\mathrm{0}} ^{\pi} \:−\int_{\mathrm{0}} ^{\pi} \:\frac{\mathrm{2}{x}}{{n}}{sin}\left({nx}\right){dx} \\ $$$$−\frac{\mathrm{2}}{{n}}\:\int_{\mathrm{0}} ^{\pi} \:{xsin}\left({nx}\right){dx}\:=−\frac{\mathrm{2}}{{n}}\left[\:\:\:\:\left[−\frac{{x}}{{n}}{cos}\left({nx}\right)\right]_{\mathrm{0}} ^{\pi} \:−\int_{\mathrm{0}} ^{\pi} −\frac{\mathrm{1}}{{n}}{cos}\left({nx}\right){dx}\right\} \\ $$$$=−\frac{\mathrm{2}}{{n}}\left\{−\frac{\pi}{{n}}\:\left(−\mathrm{1}\right)^{{n}} \:+\frac{\mathrm{1}}{{n}^{\mathrm{2}} }\left[{sinnx}\right]_{\mathrm{0}} ^{\pi} \right\}\:=\frac{\mathrm{2}\pi}{{n}^{\mathrm{2}} }\left(−\mathrm{1}\right)^{{n}} \:\Rightarrow \\ $$$${a}_{{n}} =\frac{\mathrm{2}}{\pi}×\frac{\mathrm{2}\pi}{{n}^{\mathrm{2}} }\left(−\mathrm{1}\right)^{{n}\:} \:=\frac{\mathrm{4}\pi}{\pi{n}^{\mathrm{2}} }\left(−\mathrm{1}\right)^{{n}} \\ $$$${a}_{\mathrm{0}} =\frac{\mathrm{2}}{\pi}\:\int_{\mathrm{0}} ^{\pi} \:{x}^{\mathrm{2}} \:{dx}\:=\frac{\mathrm{2}}{\pi}\left[\frac{{x}^{\mathrm{3}} }{\mathrm{3}}\right]_{\mathrm{0}} ^{\pi} \:=\frac{\mathrm{2}}{\pi}×\frac{\pi^{\mathrm{3}} }{\mathrm{3}}\:=\frac{\mathrm{2}\pi^{\mathrm{2}} }{\mathrm{3}}\:\Rightarrow \\ $$$$\bigstar{x}^{\mathrm{2}} \:=\frac{\pi^{\mathrm{2}} }{\mathrm{3}}\:+\mathrm{4}\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\frac{\left(−\mathrm{1}\right)^{{n}} }{{n}^{\mathrm{2}} }{cos}\left({nx}\right)\bigstar \\ $$$$ \\ $$

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