Banned from /sci/ so replying here.
>>>/sci/15027656
3! = 1*2*3 = the product of all elements of the set {1, 2, 3}
2! = 1*2 = the product of all elements of the set {1, 2}
1! = 1 = the product of all elements of the set {1}
0! = (*) = the product of all elements of the set {}
The product of all elements of the set {} has to be 1, not 0, and I'll prove it.
Consider that for any two DISJOINT sets A and B, product(A)*product(B) = product(union(A, B)).
Let A = {x} and B = {}.
Then product(A) = x.
Note also that in this case, union(A, B) = A.
Hence, product(A)*product(B) = product(union(A, B)) = product(A) = x.
Substitute x for product(A) and we have x*product(B) = x.
Therefore product(B) = 1.
>>>/sci/15027664
>So -1!=0!/0, so -1! is infinity?
>And -2!=-1!/-1, so -2! is -infinity?
>And -3!=-2!/-2, so -3! is infinity/2?
>retard.
Not him but no, because that's not how infinity works. 1/0 is not positive infinity. At best it's signless infinity, a quantity with infinite magnitude and undefined angle, but to make that true, you need to choose to represent the complex plane as a Riemann sphere rather than a plane. Furthermore, even if 1/0 *were* positive infinity, infinity/2 is just infinity, not infinity/2. Adding or multiplying a finite number to an infinite number just gives you the same infinite number back. Anyway, other than all that, yes, you're right: factorial is asymptotic at negative integers. See the gamma function for more info.