Question

Sol. [Key Concept: $\frac{k}{(k^{2 +}k + 1)\ ?}$ — write the term as a difference so the factorial sum telescopes.]

$$\ \sum_{k = 1}^{n}\mspace{2mu} k!\left( (k + 1)^{2} - k \right) = \sum_{k = 1}^{n}\mspace{2mu}((k + 1)(k + 1)! - k(k!))$$

$$= (n + 1)(n + 1)! - 1$$