4 years ago

Find a closed form for Sn = 1 1!+22!+...+n*n! for integer n>=1 Your response should have a factorial.

1 answer:

6 0

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}\\


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