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