According to the definition of factorial,

1 = 0!

and 0! = (-1)*0

So, the first negative integer factorial is

(-1)! = 1/0 = ∞

Again (-2)! = (-1)!/(-1) = -∞

 (-3)! = (-2)!/(-2) = (-1)!/{(-1)*(-2)} =  +∞

Generally, we see that all factorials are infinte with alternating sign,

(-n)! = (-1)!/{(-1)*(-2)*...........*(1-n)} = ∞/{(-1)*(-2)*...........*(1-n)} = (-1)ⁿ ∞