series de Fourier

x^{2} = (pi^{2}/3)+4·∑ ( ( 1/k^{2} )·cos(k·pi)·cos(kx) )
si x=pi ==> ∑ ( 1/k^{2} ) = ( pi^{2}/6 )
si x=0 ==> ∑ ( (-1)^{k}/k^{2} ) = (-1)( pi^{2}/12 )


x^{4} = (pi^{4}/5)+8pi^{2}·∑ ( ( 1/k^{2} )·cos(k·pi)·cos(kx) )+..
...+(-48)·∑ ( ( 1/k^{4} )·cos(k·pi)·cos(kx) )
si x=pi ==> ∑ ( 1/k^{4} ) = ( pi^{4}/90 )
si x=0 ==> ∑ ( (-1)^{k}/k^{4} ) = (-1)( 7pi^{4}/720 )


x^{6} = (pi^{6}/7)+12pi^{4}·∑ ( ( 1/k^{2} )·cos(k·pi)·cos(kx) )+...
...+(-240)pi^{2}·∑ ( ( 1/k^{4} )·cos(k·pi)·cos(kx) )+...
...+1440·∑ ( ( 1/k^{6} )·cos(k·pi)·cos(kx) )
si x=pi ==> ∑ ( 1/k^{6} ) = ( pi^{6}/945 )
si x=0 ==> ∑ ( (-1)^{k}/k^{6} ) = (-1)( 31pi^{4}/30240 )


x^{8} = (pi^{8}/9)+16pi^{6}·∑ ( ( 1/k^{2} )·cos(k·pi)·cos(kx) )+...
...+(-672)pi^{4}·∑ ( ( 1/k^{4} )·cos(k·pi)·cos(kx) )+...
...+13440pi^{2}·∑ ( ( 1/k^{6} )·cos(k·pi)·cos(kx) )+...
...+(-80640)·∑ ( ( 1/k^{8} )·cos(k·pi)·cos(kx) )
si x=pi ==> ∑ ( 1/k^{8} ) = ( pi^{8}/9450 )


x^{10} = (pi^{10}/11)+20pi^{8}·∑ ( ( 1/k^{2} )·cos(k·pi)·cos(kx) )+...
...+(-1440)pi^{6}·∑ ( ( 1/k^{4} )·cos(k·pi)·cos(kx) )+...
...+60480pi^{4}·∑ ( ( 1/k^{6} )·cos(k·pi)·cos(kx) )+...
...+(-1209600)pi^{2}·∑ ( ( 1/k^{8} )·cos(k·pi)·cos(kx) )+...
...+7257600·∑ ( ( 1/k^{10} )·cos(k·pi)·cos(kx) )
si x=pi ==> ∑ ( 1/k^{10} ) = ( pi^{10}/93555 )