Stirling Numbers Proof

If you use the hint and let

$$ b_k = k! \,S(n-1,k)+(k-1)!S(n-1,k-1) = a_k+a_{k-1} $$

then the series becomes

$$ \sum\limits_{k=1}^{∞} (−1)^k b_k = \sum\limits_{k=1}^{∞} (−1)^k (a_k+a_{k-1}) $$

and you will notice that when you expand the series terms will cancel each other.