考察一般项：
n/[2×3×...×(n+1)]
=n/(n+1)!
=(n+1-1)/(n+1)!
=(n+1)/(n+1)!-1/(n+1)!
=1/n!-1/(n+1)!
1/2 +2/(2×3)+3/(2×3×4)+4/(2×3×4×5)+5/(2×3×4×5×6)
=1/1!-1/2!+1/2!-1/3!+1/3!-1/4!+1/4!-1/5!+1/5!-1/6!
=1/1!-1/6!
=1 -1/720
=719/720
推广一下：
一般的：
1/2+2/(2×3)+3/(2×3×4)+...+n/[2×3×4×...×(n+1)]
=1/1!-1/2!+1/2!-1/3!+...+1/n!-1/(n+1)!
=1-1/(n+1)!