an=2/π∫(0,π)x^2cosnxdx
=2/π1/n∫(0,π)x^2dsinnx
=2/nπx^2sinnx|(0,π)-4/nπ∫(0,π)xsinnxdx
=-4/nπ∫(0,π)xsinnxdx
=4/[n^2π]∫(0,π)xdcosnx
=4/[n^2π]xcosnx|(0,π)-4/[n^2π]∫(0,π)cosnxdx
=4/n^2cosnπ-0
=(-1)^n·4/n^2
a0=2/π∫(0,π)x^2dx
=2/πx^3/3|(0,π)
=2/3π^2
所以
余弦级数为
f(x)=1/3π^2+Σ(-1)^n·4/n^2cosnx(n从1到∞)