Answer with explanation:
The given sequaence whose sum we have to find
[tex]S_{n}=1*1!+2*2!+3*3!+4*4!+...........+n*n!\\\\T_{n}{\text{general term}}=n*n!\\\\ \sum _{k=1}^{k=n}t_{k}\\\\\sum _{k=1}^{k=n}k(k!)\\\\\sum _{k=1}^{k=n}(k+1-1)(k!)\\\\\sum _{k=1}^{k=n}[(k+1)!-k!]\\\\=(1!+2!+3!+.............(n+1)!)-(1!+2!+3!+..........+n!)\\\\=(n+1)!\\\\----{\text{Cancelling out like terms}}[/tex]
→k*(k+1)=(k+1)!
