austinwang5999 austinwang5999

11-08-2016
Mathematics

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

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11-08-2016

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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