Fourier Series

n − 1 1 1 cosn 1xn 11 − cosn 1n 11 − −1 n1n 1cosn − 1xn − 101 − cosn − 1n − 1 1 − −1n−1n − 1Now the numerator in each term is the same, as −1 n1 −1 n−1 , and will be zero if n isodd and non-zero if n is even. To retain only the non-zero terms, write n 2m, b 2m 1 2n 1 2n − 18m4m 2 − 1 .Thuscosx ~ 8 ∑m18m4m 2 − 1 sin2mx13 sin2x 215sin4x 335sin6x ... .Note that we cannot match cosx at x 0,; our series must have S0 S 0, inagreement with the Dirichlet conditions.If we had sought to employ an extension in terms of an even function, then we would haveFx cosx on − x and this is already a FS, albeit with a single non-zerocoefficient.Example 2Suppose that the function fx is defined byfx x 0 x 2 x − 22 x Determine the half-range cosine series.27