求证P11+2P22+3P33+……=nPnn=P(n+1)(n+1)-1

由Pkk=k!得
P11+2P22+3P33+……+nPnn
=1*1!+2*2!+3*3!+……+n*n!
=(2-1)1!+(3-1)*2!+(4-1)*3!+……+((n+1)-1)*n!
=(2!-1!)+(3!-2!)+(4!-3!)+……+((n+1)!-n!)
=(n+1)!-1=P(n+1)(n+1)-1
一楼回答基本正确，但第4行中(3-2)*2!应改为(3-1)*2!

由Pkk=k!得
P11+2P22+3P33+……+nPnn
=1*1!+2*2!+3*3!+……+n*n!
=(2-1)1!+(3-2)*2!+(4-1)*3!+……+((n+1)-1)*n!
=(2!-1!)+(3!-2!)+(4!-3!)+……+((n+1)!-n!)
=(n+1)!-1=P(n+1)(n+1)-1