Fourier Series

fx dx 0 2 dx 1 0x − 2 dxa 0 1 − 1 2 0x −1 1 −13 x − 3 0423a n 1 −fxcosnx dx01 − 0 1 1 2 n 2 2 cosnx dx 1 0x − 2 cosnx dx1n x − 2 sinnx − 10 02n x − cosnx − 1 2 0 02n x − sinnx dx2 cosnx dx2nIn a similar manner, determine b n asb n n cosn − 2n 3 1 − cosn n −1n − 2n 3 1 − −1n Thus the final series isfx ~ Sx 223∑n12n cosnx 2 n −1 n − 2n 1 − 3 −1n sinnx .The series can be evaluated at x 0, by using the Dirichlet Conditions.(i) At x 0: the function is continuous and fx 2 . Thus 2 223∑n12 ∑n 2n11n 2 26 .(ii) At x : the function is discontinuous with fx 0 2 and fx − 0 0, so theseries converges to 1 2 fx 0 fx − 0 2 /2. Thus 22223∑n12−1nn 2 ∑n1−1n1n 2 212 .21