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.

Answer 1

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

Answer 2

clever telescoping