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.

11 Responses to “Factorials and Exponentials.”

1. bubbloy said

What are you trying to figure out about ((n!)!)! ?

• fretty said

His conjecture is that if you have m nested exponentials on the top and m nested factorials on the bottom, then the limit as n tends to infinity is 0.

Maybe try using logs…both the exponentials and the factorials will turn into products and sums then.

Maybe try using the power series for e^x too (although I am not sure that this will work too well). Induction might give you the result after one application of this.

I dunno, just first thoughts really.

2. You can do this by induction on the number of factorials and by using the definition of a limit carefully.

3. Anonymous said

Show using induction and Stirling that ln(ln(… (ln(n!!…!))…) > 0.5 n ln(n) for large enough n, where there the same number of ln’s as !’s. I think that n >= 8 should work.
But ln(ln(…(e^e^…^n))…) = n, where there are the same number of e’s as ln’s.
Now [0.5 n ln(n)]/n –> infinity and [0.5 n ln(n)]/e^n –> 0.

• James said

This is a nice way to go. And it’s probably easy to adapt when there’s not the same number of e’s and !’s. I’ll try this out.

• You don’t need any explicit bounds if you just want to show that the expression converges to 0. The following is schematic: if, for strictly increasing f and g, for any E>0 there is an N such that f(n)/g(n) N, then certainly e^(f(n))/g(n)! N. Thus the result you want follows by induction on the number of factorials/exponents.

• My previous message got messed up when I posted it, for some reason…

• James said

I’m not exactly sure how your last line followed, but for the estimate it gives a good lower-bound to some of these. For the ones which go to infinity, this is nice, but for the ones where the estimate does not go to infinity, this doesn’t tell me all that much. Nonetheless, this is a pretty sweet estimate.

4. Good article. I will be facing many of these issues as well..

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