For Dominick only! Consider the Gamma function Γ(z) defined by the integral Γ(z) = ∫₀^∞ t^z-1 e^-t dt.   See also Digital Library of Math functions.

1. Show that Γ(1)=1 (easy!)
2. Use integration by parts to show that Γ(z+1)=z Γ(z) (not too hard, Math 221 level)
3. Deduce that Γ(n+1)=n! for integer n (easy from (1) and (2)).
4. Deduce that 0! = 1 indeed. (Woa! cool. There's more to 0!=1 then a mere convention after all!)
5. Show that Γ(0), Γ(-1), Γ(-2), ... diverge! (hence "(-1)!" = ∞, "(-2)!" = ∞, etc... ).
6. Note the historical annoyance: why didn't they define Γ(z) as ∫₀^∞ t^z e^-t dt, then we'd have Γ(n) = n! instead of Γ(n+1) = n!. Gauss tried.