serie zeta alternada

teorema:
∑ ( (-1)^{n}/n^{2s} ) =  (-1)^{s}·pi^{2·s}·( 2^{2·s+(-1)}+(-1) )·B_{2s+(-1)}(0)


∑ ( (-1)^{n}/n^{2} ) =  (-1)·(pi^{2}/12)
∑ ( (-1)^{n}/n^{2} ) =  (-1)^{1}·pi^{2·1}·(2^{2·1+(-1)}+(-1))·(1/12)
∑ ( (-1)^{n}/n^{4} ) =  (-1)·(7pi^{4}/720)
∑ ( (-1)^{n}/n^{4} ) =  (-1)^{2}·pi^{2·2}·(2^{2·2+(-1)}+(-1))·(-1)(1/720)
∑ ( (-1)^{n}/n^{6} ) =  (-1)·(31pi^{6}/30240)
∑ ( (-1)^{n}/n^{6} ) =  (-1)^{3}·pi^{2·3}·(2^{2·3+(-1)}+(-1))·(1/30240)
∑ ( (-1)^{n}/n^{8} ) = (-1)(127pi^{8}/1209600)
∑ ( (-1)^{n}/n^{8} ) =  (-1)^{4}·pi^{2·4}·( 2^{2·4+(-1)}+(-1) )·(-1)(1/1209600)
∑ ( (-1)^{n}/n^{10} ) = (-1)(511pi^{10}/47900160)
∑ ( (-1)^{n}/n^{10} ) =  (-1)^{5}·pi^{2·5}·( 2^{2·5+(-1)}+(-1) )·(1/47900160)




teorema:
∑ ( (-1)^{n+1}/n^{2s} ) =  ( 1+( (-1)/2^{2s+(-1)} ) )·∑ ( 1/n^{2s} )




demostració:
(-1)∑ ( (-1)^{n}/n^{2s} ) =  (-1)^{s+1}·pi^{2·s}·( 2^{2·s+(-1)}+(-1) )·B_{2s+(-1)}(0)


(-1)∑ ( (-1)^{n}/n^{2s} ) =  (-1)^{s+1}·pi^{2·s}·( 2^{2·s+(-1)}·B_{2s+(-1)}(0)+...
...+(-1)^{s}·pi^{2·s}B_{2s+(-1)}


(-1)∑ ( (-1)^{n}/n^{2s} ) =  ∑ ( 1/n^{2s} )+(-1)^{s}·pi^{2·s}B_{2s+(-1)}


(-1)∑ ( (-1)^{n}/n^{2s} ) =  ∑ ( 1/n^{2s} )+...
...+( (-1)/2^{2s+(-1)} )·(-1)^{s+1}·pi^{2·s}·2^{2s+(-1)}B_{2s+(-1)}




(-1)∑ ( (-1)^{n}/n^{2s} ) =  ∑ ( 1/n^{2s} )+( (-1)/2^{2s+(-1)} )·∑ ( 1/n^{2s} )


(-1)∑ ( (-1)^{n}/n^{2s} ) =  ( 1+( (-1)/2^{2s+(-1)} ) )·∑ ( 1/n^{2s} )