1÷(x+1)(x+3)+1÷(x+3)(x+5)+1÷(x+5)(x+7)+1÷(x+7)(x+9)+……+1÷(x+2n-1)(x+2n+1)
=1/(x+1)(x+3)+1/(x+3)(x+5)+1/(x+5)(x+7)+1/(x+7)(x+9)+……+1/(x+2n-1)(x+2n+1)
=[1/(x+1)-1/(x+3)]/2+[1/(x+3)-1/(x+5)]/2+[1/(x+5)-1/(x+7)]/2+……+[1/(x+2n-1)-1/(x+2n+1)]/2
=1/[2(x+1)]-1/[2(x+3)]+1/[2(x+3)]-1/[2(x+5)]+1/[2(x+5)]-1/[2(x+7)]+……+1/[2(x+2n-1)]-1/[2(x+2n+1)]
=1/[2(x+1)]-1/[2(x+2n+1)]
=(x+2n+1-x-1)/[2(x+1)(x+2n+1)]
=n/[(x+1)(x+2n+1)]
楼主追问“我不是他舅”先生：一直加到101,结果是?
楼主这里说的101,应该是一直加到1÷/(x+101)(x+103)吧?
套用上面我推导出来的公式,应该是n=51
有：
1÷(x+1)(x+3)+1÷(x+3)(x+5)+1÷(x+5)(x+7)+1÷(x+7)(x+9)+……+1÷(x+101)(x+103)
=51/[(x+1)(x+2×51+1)]
=51/[(x+1)(x+103)]
要是还想继续算的话,就是把分母写成多项式的形式.那么,楼主所求的结果就是：
51/(x^2+104x+103)