Question

Find a closed form for

Answer 1

$S_n=1\cdot1!+2\cdot2!+\cdots+n\cdot n!$
$S_n=(1+1-1)\cdot1!+(2+1-1)\cdot2!+\cdots+(n+1-1)\cdot n!$
$S_n=(2\cdot1!-1\cdot1!)+(3\cdot2!-1\cdot2!)+\cdots+\bigg((n+1)n!-\cdot n!\bigg)$
$S_n=(2!-1!)+(3!-2!)+\cdots+\bigg((n+1)!-1\cdot n!\bigg)$
$S_n=-1!+(n+1)!$
$S_n=(n+1)!-1$