4/(x+1)(x+3)+4/(x+3)(x+5)+.+4/(x+2n-1)(x+2n+1)
=2(1/x+1-1/x+3+1/x+3-1/x+5+.+1/x+2n-1-1/x+2n+1)
=2(1/x+1-1/x+2n+1)
=2(2n/(x+1)(x+2n+1))
=4n/(x+1)(x+2n+1)
算到这应该可以了吧