Quick Limit question

  • Thread starter Zurtex
  • Start date
  • #1
Zurtex
Science Advisor
Homework Helper
1,120
1
I'm doing some tutoring tomorrow morning for a girl who needs to do some maths resits. I was looking through the papers and you need to prove that:

[tex]n^{n-1} \geq n! \quad \forall n > 1 \; \text{and} \; n \in \mathbb{N}[/tex]

Which is fine, but it asks to you to do it by proving by definition that:

[tex]\lim_{n \rightarrow \infty} \frac{n!}{n^n} = 0[/tex]

And I have to admit that I can't quite remember how to do this one, if someone could point me in the right direction that would be great, I remember seeing a proof for it so I'm sure it'll come back to me.
 

Answers and Replies

  • #2
Hurkyl
Staff Emeritus
Science Advisor
Gold Member
14,967
19
Split it into a part that obviously goes to zero, and a part that is obviously bounded!
 
  • #4
Zurtex
Science Advisor
Homework Helper
1,120
1
iNCREDiBLE said:
Too far ahead for the material she is expected to know and my brain isn't working at the moment, I don't know what you mean Hurkyl
 
  • #5
TD
Homework Helper
1,022
0
I doubt this would pass as a proof, but here 1/n goes to zero and the second limit has an equal number of factors in the nominator as denominator. The highest factors are equal (n) but in the nominator, they decrease while they don't in the denominator.

[tex]\mathop {\lim }\limits_{n \to \infty } \frac{{n!}}{{n^n }} = \mathop {\lim }\limits_{n \to \infty } \frac{{n!}}{{n \cdot n^{n - 1} }} = \mathop {\lim }\limits_{n \to \infty } \frac{1}{n}\mathop {\lim }\limits_{n \to \infty } \frac{{n!}}{{n^{n - 1} }} = \mathop {\lim }\limits_{n \to \infty } \frac{1}{n}\mathop {\lim }\limits_{n \to \infty } \frac{{\overbrace {n \cdot \left( {n - 1} \right) \cdot \ldots \cdot 2}^{n - 1}}}{{\underbrace {n \cdot n \cdot \ldots \cdot n}_{n - 1}}}=0[/tex]
 
  • #6
George Jones
Staff Emeritus
Science Advisor
Gold Member
7,599
1,477
[tex]
\left( \frac{n \cdot (n-1) ... 2} {n \cdot n ... n} \right) \frac{1} {n}
[/tex]

The stuff in the big brackets is less greater than zero and less than 1 for n > 2.

Regards,
George
 

Suggested for: Quick Limit question

  • Last Post
Replies
29
Views
499
  • Last Post
Replies
3
Views
576
  • Last Post
Replies
7
Views
522
  • Last Post
Replies
11
Views
582
Replies
2
Views
96
  • Last Post
Replies
8
Views
394
  • Last Post
Replies
3
Views
404
  • Last Post
Replies
3
Views
607
  • Last Post
Replies
1
Views
438
  • Last Post
Replies
5
Views
594
Top