出来了,我一会写好详细过程.
我算来还是比较麻烦的,总感觉应该有更简单的方法,无奈暂时没想到；
令Cn=Bn*B(n+1)
令
S=b1b2-b2b3+……+(-1)^n*bn*b(n+1)
=C1-C2+C3-C4+...+(-1)^(n+1)*Bn*B(n+1)
=3/3 * 5/3 - 5/3 * 7/3 + 7/3 * 9/3 - 9/3 * 11/3 +...+(-1)^(n+1)*(2n+1)/3 * (2(n+1)+1)/3
9S=3*5-5*7+7*9-9*11+...+(-1)^(n+1) * (2n+1)*(2n+3)
n为奇数时：
9S=5*(3-7)+9(7-11)+...+ (2n+1)*(2n+3)
(9S - (2n+1)*(2n+3))/(-4)=5+9+13+...(2n-1) = 4*(n+2)
S=(1/9)*[ (2n+1)*(2n+3) - 16*(n+2)]
n为偶数时：
9S=5*(3-7)+9(7-11)+...- (2n+1)*(2n+3)
9S/(-4)=5+9+13+...+2n+1 = 4*(n+3)
S=(-16/9)*(n+3)