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 \lim_{n\to\infty}\dfrac{e^n}{n!} = 0.  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:

\lim_{n\to\infty}\dfrac{e^{e^n}}{n!}.

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

\lim_{n\to\infty}\dfrac{e^{e^n}}{(n!)!}

where this double factorial is just n! with another factorial at the end.  It turns out, this one goes to 0. 

 

The problem here is that after ((n!)!)!, 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 ((n!)!)! 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 f\sim g then f\circ f \sim g\circ g under such-and-such a condition.

 

Feel free to comment below if you think of anything.

EDIT: It seems that scribd is now behind a paywall now.  :(  

Brown and Churchill (8th ed) was the book I used for the second complex analysis class I’ve had to take so far (the first was Lang).  My class went over the first six chapters and half of the seventh: so, up to the middle of the section on applications of residues.

To prep for the final, I compiled a quick, slightly-shorter list of things that I feel the complex student should know if they’ve used this book and have gotten to around the same point.  I’ve excluded the chapter on applications of residues, since it’s a relatively short chapter with better pictures in the text than ones I could draw at 5am.  Because sharing is caring, below is a link to the pdf.  Enjoy!

http://www.scribd.com/doc/45170305

So, I wanted to proceed onwards towards some pretty cool mathematics (and, finally, get through basic linear algebra) by introducing the Gram-Schmidt ortho-normalization process and some really sweet consequences (that actually really surprised me!), but it occurred to me that I’d need to introduce norms and inner products as well as prove a butt-load of things about them.

Because I am terribly lazy, I am not going to do this.  Instead, I read through a number of inner product introductions (which are all basically the same) to find one that was well-written.  The one that I’ve picked to show ya’ll is from G. Keady, from the University of Birmingham.  It is in pdf form, and it is available here (warning, pdf!).

With the possible exception of ultra-brevity (orthonormal is abbreviated ON, and that’s kind of weird to get used to) and some of the things at the end of the paper, this is a 3-page introduction and, partially because of this, is very readable.  You should not need any math besides what we’ve already covered in this blog.

We’ll give examples of normed vector spaces and inner product spaces later, but we’ll definitely be using the inner product space of continuous (real) polynomials on the interval [0,1], which has the inner product

\displaystyle\langle f, g\rangle = \int_{0}^{1} f(x)g(x)dx

This will come in handy later, so remember it!

I’m not going to write a whole huge thing on here since, fortunately, I started a big paper on sets to teach my old students.  I didn’t finish it entirely, or proofread it entirely, but I’m going to post and repost until it’s completely done.  But there’s no sense in keeping it from you; just start it now, and by the time you get up to the end, I’ll hopefully be done with the rest!

Due to the formatting issues (desktop latex just uses stuff that looks like $ this $, but wordpress formats statements that look like $.latex this$ (without the period) and changing between the two would take far too long), I’ve decided just to upload the pdf to this blog.

Go here for the Set Theory guide.