Question

(1)计算：1!+2*2!+3*3!+……+n*n!=？ (2)求证：k/(k+1)!=1/k!-1/(k+1)! (3)求证：1/2!+2/3!+n/(n+1)!...

(1)计算：1!+2*2!+3*3!+……+n*n!=？
(2)求证：k/(k+1)!=1/k!-1/(k+1)!
(3)求证：1/2!+2/3!+n/(n+1)!=1-1/(n+1)

Answer 1

(1)
ak=k*k!=(k+1-1)k!=(k+1)!-k!
1!+2*2!+3*3!+...+n*n!
=1!+3!-2!+4!-3!+...+(n+1)!-n!
=(n+1)!-1
(2)
1/k!-1/(k+1)!=1/k!-1/[(k+1)k!]
=(1/k!)[1-1/(k+1)]
=k/(k+1)! (3)限制长度了，提示用第二问结论。