化简两个题目 高二数学

(注意下面带***的两行形式上的相似性)
(1)设结1/2!+2/3!+3/4!+····+(n-1)/n!=S
则S+1/n!=1/2!+2/3!+3/4!+·······+(n-1)/n!+1/n! ***
=1/2!+2/3!+3/4!+····+(n-2)/(n-1)!+n/n!
=1/2!+2/3!+3/4!+····+(n-2)/(n-1)!+1/(n-1)! ***
=....=1/2!+1/2!=1
所以S=1-1/n!
(2)设1*1!+2*2!+3*3!+·····+n*n!=S
则S-(n+1)!=1*1!+2*2!+3*3!+·····+n*n!-(n+1)! ***
=1*1!+2*2!+3*3!+·····+n*n!-(n+1)n!
=1*1!+2*2!+3*3!+·····+(n-(n+1))n!
=1*1!+2*2!+3*3!+·····+(n-1)(n-1)!-n! ***
=...=1*1!-2!=-1
所以S=(n+1)!-1

错位相消。。。20分也叫重谢？？？？？？？
第一题。。你＋1除2！＋1除3！＋。。。＋1除n！就行了
第二题。。你＋1！＋2！＋。。。＋n！就行了