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.

Question

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.

ganeshie8 · Answer

$$\begin{align}S_n &= \color{red}{1} \cdot 1! +  \color{red}{2} \cdot 2! + \ldots +  \color{red}{n} \cdot n!\~\
&=  \color{red}{(2-1)} \cdot 1! +  \color{red}{(3-1)} \cdot 2! + \ldots +  \color{red}{(n+1-1)} \cdot n!\~\
&=2!-1! + 3!-2! +\cdots + (n+1)!-n!\~\
&=-1! + (n+1)!
\end{align}$$

anonymous · Answer

You might recall asking this about a month ago... http://openstudy.com/users/zmudz#/updates/55ad142ee4b0d48ca8ed35d0