Value of $3-\sum_{k=2}^\infty\frac{1}{(k-1) \cdot k \cdot k!} $

Question

I am trying to find the limit of the sequence $a_n$ defined by the initial term $a_1=3$ and the recurrence relation: $$a_n = a_{n-1} - \frac{1}{(n-1) \cdot n \cdot n!}, \quad \text{for } n = 2, 3, \ldots.$$I need to determine: $$\lim_{n \to \infty} a_n$$ I started writing out terms and got $$a_n = 3-\sum_{k=2}^{n} \frac{1}{(k-1) \cdot k \cdot k!} $$ which is almost telescopic but that $k!$ is in the way. I don't know how to proceed from here so aby help would be appreciated.