Question

S=1×1!+2×2!+3×3!+……+n×n!

则S=

Answer 1

一、1x1!+2x2!+3x3!+4x4!+.....+nxn!=(n+1)!-1
证明：
左边=(2-1)x1!+2x2!+3x3!+4x4!+.....+nxn!
=2!-1+2x2!+3x3!+4x4!+.....+nxn!
=(1+2)x2!-1+3x3!+4x4!+.....+nxn!
=3!-1+3x3!+4x4!+.....+nxn!
=(1+3)x3!-1+4x4!+.....+nxn!
=4!-1+5x5!+.....+nxn!
..........
=(n+1)!-1
所以：S=1×1!+2×2!+3×3!+……+n×n!=(n+1)!-1

Answer 2

n×n!=(n+1-1)n!=(n+1)!-n!

S=1×1!+2×2!+3×3!+……+n×n!
=2!-1!+3!-2!+……+(n+1)!-n!
=(n+1)!-1