Find the Fourier Series for f(x) =1 on the interval 0<= x <= pi in terms of [itex] \phi_{n} = \sin{nx} [/itex]. Bu integrating this series find a convergent series for the function g(x) =x on this interval assuming the set {sin nx} is complete(adsbygoogle = window.adsbygoogle || []).push({});

Ok for the FOurier Coefficients

[tex] c_{n} = \frac{\int_{0}^{\pi} f \phi \rho dx}{\int_{0}^{\pi} \phi^2} dx [/tex]

this is how it is in my test

rho is suppsod to be the weight

ok for the numerator

[tex] \int_{0}^{\pi} \sin{nx} dx = \frac{1}{n} [- \cos{nx}]_{0}^{\pi} = (-1)^n + 1 [/tex]

for hte denominator

[tex] \int_{0}^{\pi} (\sin{nx})^2 dx = \frac{\pi}{2} - \frac{\sin{2n \pi}}{4} =\frac{\pi}{2} [/tex]

so the fourier series is

[tex] \frac{2}{\pi} \sum_{n=1}^{\infty} \frac{\sin{nx}}{n} (-1^n + 1) [/tex]

what do they mean by integrate the series? Does it mena i should integrate the argument of this sum? ANd how would one find a convergent series for the function g(x) =x??

Please help!

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# Homework Help: Differential Equations

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