n/(n+1)!=1/n!-1/(n+1)!,(1/2的阶乘+2/3的阶乘+。。。+n/(n+1)的阶乘)=1/n!-1/(n+1)!+1/(n-1)!-1/n!+...+1/2!-1/3!+1/1!-1/2!=1-1/(n+1)!故(1/2的阶乘+2/3的阶乘+。。。+n/(n+1)的阶乘)的极限为1.