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,

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