a) [tex]1\cdot 1! + 2\cdot 2! + \dots + n\cdot n! = \sum_{k=1}^{n} k\cdot (k)!= \sum_{k=1}^{n} (k+1-1) \cdot (k)! = \sum_{k=1}^{n} [(k+1)!-k!] = 2! + 3! + \dots + n! + (n+1)! - 1!-2!-3!-\dots -n! = (n+1)!-1[/tex]
b) [tex] \dfrac{1}{2!} + \dfrac{2}{3!} +\dots +\dfrac{n}{(n+1)!} = \sum_{k=1}^{n} \dfrac{k}{(k+1)!} = \sum_{k=1}^{n} \dfrac{k+1-1}{(k+1)!} = \sum_{k=1}^{n} [\dfrac{1}{(k)!} - \dfrac{1}{(k+1)!} = 1-\dfrac{1}{(n+1)!}[/tex]
