Squeeze Theorem with limits n/n as n approaches 0

[ReginaldN]

Homework Statement

The question asks to use the squeeze theorem to show that the limit of n!/n^n equals 0 as n approaches ∞.

Homework Equations

I need to use the squeeze theorem to solve this problem, and I'm not sure what the upper limit is.

The Attempt at a Solution

I found the lower limit to be 1/n^n, and I don't remember how I got here. I need some help.

 

[ReginaldN]

Isn't it just n^1? Which is just n?

 

[Dick]

You can write n!/n^n as (1/n)*(2/n)*(3/n)*...*((n-1)/n)*(n/n), right? All of those factors are less than or equal to 1. How many of those factors are less than or equal to 1/2?

 

[ReginaldN]

hmm, isn't 2/2^2 equal to 1/2? So only one of those is less than or equal to 1/2?

 

[Dick]

hmm, isn't 2/2^2 equal to 1/2? So only one of those is less than or equal to 1/2?

No, take n=6. That's 6!/6^6=(1/6)*(2/6)*(3/6)*(4/6)*(5/6)*(6/6). Isn't it? I think three of those are less than or equal to 1/2. The first three. Can you generalize? n odd is a little different, but don't worry about that right now.

 

[ReginaldN]

okay so, for n=6. everything less than 3/6 is less than or equeal to one half? I'm not really understanding what you mean by generalize.

 

[Dick]

okay so, for n=6. everything less than 3/6 is less than or equeal to one half? I'm not really understanding what you mean by generalize.

Generalize means how many for n=10, n=20 etc. Now how many for a general even value of n?

 

[ReginaldN]

So for about half of the values of n, half of them will be great than or equal to 1/2, right?

 

[deluks917]

Can you show n!/(nⁿ) ≤ (1/n) ?

 

[Dick]

Can you show n!/(nⁿ) ≤ (1/n) ?

deluks917 is right and has a much simpler approach. I was trying to get you to show n!/n^n<=(1/2)^(n/2). That works but you don't need that much unless you are trying to show the series converges. n!/n^n<1/n will do just fine. Can you say why that must be true?

 

[ReginaldN]

that must be true, because no matter what integer n is, it will always end up as 1/n. For example, if n = 2 , then the equation would be 2/2², which is 2¹ / 2², and that would equal 1/2¹.