## zmudz one year ago Find a closed form for $$S_n = 1 \cdot 1! + 2 \cdot 2! + \ldots + n \cdot n!.$$ for integer $$n \geq 1.$$ Your response should have a factorial.

1. anonymous

Notice that $n\cdot n!=(n+1-1)\cdot n!=(n+1)\cdot n!-n!=(n+1)!-n!$ Now, \begin{align*}S_n&=1\cdot1!+2\cdot2!+\cdots+(n-1)\cdot(n-1)!+n\cdot n!\\[1ex] &=(2!-1!)+(3!-2!)+\cdots+(n!-(n-1)!)+((n+1)!-n!)\\[1ex] &=\cdots \end{align*}

2. anonymous

clever telescoping