+26367 Find a closed form for the sum \(S = 1*1! + 2*2! + 3*3! + ... + n*n!\).
\(\begin{array}{|rcll|} \hline n*n! &=& (n+1-1)*n! \\ &=& \Big((n+1)-1\Big)*n! \\ &=& (n+1)*n!-1*n! \\ \mathbf{n*n!} &=& \mathbf{(n+1)!-n!} \\ \hline \end{array}\)
Telescoping sum
\(\begin{array}{|rcll|} \hline 1*1! &=& 2!-1! \\ 2*2! &=& 3!-2! \\ 3*3! &=& 4!-3! \\ 4*4! &=& 5!-4! \\ 5*5! &=& 6!-5! \\ \ldots && \ldots \\ \mathbf{n*n!} &=& \mathbf{(n+1)!-n!} \\ \hline \\ \text{S} &=& -1! +(n+1)! \\ \mathbf{S} &=& \mathbf{(n+1)! - 1} \\\\ \hline \end{array}\)
\(S = 1*1! + 2*2! + 3*3! + ... + n*n! = \mathbf{(n+1)! - 1}\)
