Equation with Factorials

One of my Patrons shared this with me, I thought it was neat. I do not know where it came from.

[OilRig-]

A video about negative factorials would be exciting

[rowrow_]

Andy had to go but he came back super quick

[leahithink]

okay, now we need a follow-up video about the gamma function

[FurbleBurble]

I don't know anything about negative factorials, so I think a video on that would be interesting. Also, and I've said this before but I want to say it again, I really appreciate that you go over all the steps and the relevant rules and theorems and what not. Thank you for providing these videos.

[fortnitevideo]

Please make a video about negative factorials

[drakkondarkspell]

A video about negative factorials? How exciting!

[GlorifiedTruth]

Oh, man, I got this one in a heartbeat. Finally!

[anorris]

Quicker is to multiply the first term by p/p in the first step rather than all terms by p!. You bring the first term to the same denominator as the others this way, which is more intuitive.

[pie]

A negative factorials video would be interesting!

[rupom_]

Andy yes please tell us about negative factorials
(Also can you explain why the 'x!' in desmos has a graph for all real values of x)

[paulvansommeren]

2:30 i'm pretty sure Andy really had to p

[jakubl]

3:10 - you cancelled 4! only to get it back as a common denominator few seconds later.

[EthanEthan-mloj]

well using the knownledge about positive integers it seems to be every multiple of a said integer, also it might be similar to when a negative number is raised to a positive number ex: -5^2 = -25 and (-5)^2 = 25 so it would be like -5! = -120 and (-5)! = 120

[matesafranka]

As I understand, there is an extension of the factorial function that has a broader domain, but it only includes noninteger negatives, so it still wouldn't help here.

[Nicoya]

Negative integer factorials are a little tricky. They are, by all reasonable definitions, undefined. However, you can, under certain circumstances, manipulate them algebraically to get a sensible, defined answer. For example if you take n<0 then you can take n! = ∏k where k = n to -∞, which is of course undefined. But if you then have a!/b! where a < 0 and b < 0 then if a < b, a!/b! = 1/((a+1)(a+2)...(b)) and if b < a, a!/b! = ((b+1)(b+2)...(a)) and if a=b then a!/b! = 1, all of which are defined. With a little mathematical handwaving, this gives us the result we'd expect from the definition where n! = n * (n-1)!.

Obviously there's a lot of caveats that come along with this since we're cancelling infinite numbers of terms from a non-convergent product, so you shouldn't just blindly apply it unless you really know what you're doing.

[QuillPGall]

i’d love some negative factorial stuff. anything that fucks around with the gamma function is fire

[Grassmpl]

Fatorial is undefined for negative integers. For general complex numbers use the gamma function.

[ИскандерЖалилов]

It is fun to see the graph of this equation, it has intersections with X axis on every negative integers but -3

[petrouchkita]

how did you came back that quickly😮😮, btw a guy named 'Lines that connects' made a video abt factorials

[reeb]

finally some easy mental math questions