How to find the summation of this infinite series: $sum_{k=1}^{infty} frac{1}{(k+1)(k-1)!}(1 - frac{2}{k})$

I've been trying to figure out the following sum for a while now:

$$\sum_{k=1}^{\infty} \frac{1}{(k+1)(k-1)!}\left(1 - \frac{2}{k}\right)$$

I'm pretty sure that this doesn't evaluate to $0$.

As $k$ increases the term tends to $0$, but the first few terms add up to give a non-zero number.

I'm just having trouble figuring out how to find that number. Any help would be appreciated.