Question

Find a closed form for Sn = 1 *1!+2*2!+...+n*n! for integer n>=1 Your response should have a factorial.

Answer 1

Hello,


`x_(1) =1*1!=1\\ x_(2) =2*2!=4\\ x_(3) =3*3!=18\\ x_(4) =4*4!=96\\ S_(1) = x_(1)=1=2!-1\\ S_(2) = x_(1)+x_(2)=1+4=5=3!-1\\ S_(3) = x_(1)+x_(2)+x_(3)=5+18=23=4!-1\\ S_(4) = x_(1)+x_(2)+x_(3)+x_(4)=23+96=119=5!-1\\ \boxed{S_(n) = (n+1)!-1}\\`