[tex]1\cdot 1!+2\cdot 2!+...+100\cdot 100! = \\ \\ = \sum\limits_{k=1}^{100}(k\cdot k!)=\sum\limits_{k=1}^{100}\Big[(k+1-1)\cdot k!\Big] = \\ \\ = \sum\limits_{k=1}^{100}\Big[(k+1)\cdot k!-k!\Big] = \sum\limits_{k=1}^{100}\Big[(k+1)!-k!\Big] = \\ \\ = 2!+3!+4!+...+101! - 1!-2!-3!-...-100! = \\ \\ = 101!-1! = \boxed{101!-1}[/tex]
