Factorials and Exponentials.
November 6, 2011
I’ve been working on a problem (here is a partial paper with some ideas) that’s really easy for any calculus student to understand but quite difficult for even wolfram alpha to work out some cases. Here’s the idea:
We know that . It’s not hard to reason this out (there are some relatively obvious inequalities, etc.), but I wanted to know what happened if we considered something like:
It turns out, this goes to infinity. Maybe this is not so surprising. But, to balance this out, I thought maybe I could add another factorial on the bottom. What about
where this double factorial is just with another factorial at the end. It turns out, this one goes to 0.
The problem here is that after , Mathematica doesn’t seem to be able to handle the sheer size of these numbers. Consequently, I only have a few values for this. I’ve included everything I have in a google-doc PDF (the only way I can think to share this PDF), and I’m looking for suggestions. Here’s some things I thought of:
- Stirling’s formula. Unfortunately, this starts to get very complicated very quickly, and if you consider subbing it in for even it can take up a good page of notes. It also doesn’t reduce as nicely as I’d like.
- Considering the Gamma function. It may be easier to work with compositions of the gamma function since it is not discrete and we may be able to use some sort of calculus-type things on it.
- Number Crunching. For each of these cases, it seems like there is a point where either the numerator or the denominator "clearly" trumps over the other; this is not the "best" method to use, but it will give me some idea of which values potentially go to infinity and which go to zero.
- Asymptotics. I’m not so good at discrete math or asymptotics, so there may be some nice theorems (using convexity, maybe?) in that field that I’ve just never seen before. Especially things like: if then under such-and-such a condition.
Feel free to comment below if you think of anything.