# Limit as n->infinity of (n^n)/n!

1. May 18, 2015

### Potatochip911

1. The problem statement, all variables and given/known data
Evaluate the limit $\lim_{n\to\infty} \dfrac{n^n}{n!}$

2. Relevant equations
Ratio test: $a_{n+1}*\dfrac{1}{a_n}$
3. The attempt at a solution
I was having trouble evaluating this so I tried to use the ratio test which unfortunately leads to $\lim_{n\to\infty} \dfrac{n}{n+1}$ which just ends up being 1 which does not tell me whether the series is convergent or divergent. Alternatively I tried looking at it like this but I couldn't see where to go from there $$\lim_{n\to\infty} \frac{n}{1} \frac {n}{2} \frac {n}{3} \cdots \frac{n}{n}$$

2. May 18, 2015

### Orodruin

Staff Emeritus
You can use either the last expression that you gave to bound the expression from below by a diverging function, or you can take the logarithm of everything and apply Stirling's approximation.

3. May 18, 2015

### Potatochip911

Sorry I'm quite lost as to how I would go about finding a boundary for this function, is there a theorem or something I should be aware of?

4. May 18, 2015

### Noctisdark

Hi potatochip911,
n^n/n! = (n*n*....n)/(1*2*....n)
(n+1)^(n+1)/(n+1)! = (n+1)(n+1)...../(1*2*...(n+1)),
Hope this makes it easier, you can take out (n+1) from the second and compare the two series, and use induction, if the n^n/n! Is divergent then so is (n+1)^(n+1)/(n+1)!

5. May 18, 2015

### Orodruin

Staff Emeritus
How about n/k > 1 for n > k?

6. May 18, 2015

### Potatochip911

I don't think I'm allowed to use induction to solve these problems.

Can I just do this? $$\frac{n^2}{2} < \frac{n^n}{n!} \mbox{on the interval} [1,\infty] \\ \lim_{n\to\infty} \frac{n^2}{2}=\infty$$ and then since a function smaller function diverges the original must also diverge? Edit: I think I messed this up again I keep thinking in terms of sums

7. May 18, 2015

### Orodruin

Staff Emeritus
You can even do this: $n = n/1 < n^n/n!$

8. May 18, 2015

### PeroK

You don't need induction, although it's perfectly valid to use induction for any maths problem - unless it's explicitly stated that you can't. Anyway, back to your problem. Where did you get $n^2/2$ from?

$n^n$ and $n!$ both have $n$ terms; $n^n$ is clearly much larger than $n!$ as $n$ increases. So, all you need is a crude estimate. There is, in fact, a very simple estimate. Can you spot it?

PS: I see Orodruin beat me to it!

PPS: You are also allowed to calculate some values. For example, if you tried n = 5, you'd see that:

$5! = 120$ whereas $5^5 = 25 \times 125$

So, it should be fairly obvious from that where this sequence is heading!

Last edited: May 18, 2015
9. May 18, 2015

### Orodruin

Staff Emeritus
I would say he used the first two factors of:
rather than just the first.

Edit: Of course, this does not tell you how fast it diverges. Stirling's approximation will do that for you (and you should find that it diverges exponentially).

10. May 18, 2015

### Potatochip911

Okay I will keep that in mind for the future.

Yea its just I'm going through a textbook from start to finish and we haven't touched on induction yet so I didn't want to use it. Thanks for the help guys!