Proof of lim (n!)^(1/n)

  • Thread starter foo_daemon
  • Start date
  • #1

Homework Statement


Prove the [tex]\lim_{n\to +\infty}{\sqrt[n]{ n! }} \equiv \infty [/tex]


Homework Equations


uses well-known operations

The Attempt at a Solution


I think the best (easiest) approach is to find some [tex] f(n) \leq n! [/tex] , and express it as some [tex]g(n)^{n}[/tex] . This will then get rid of the annoying n-root, and it should then be easy to show that [tex]\lim_{n\to +\infty}{g(n)} = \infty [/tex] , which implies the limit is infinity for [tex]n![/tex] (i.e. the squeeze theorem only with regard to the lower bound).

Since [tex]n! = (n)(n-1)(n-2) ... (n-n+2) [/tex] , I thought of using the smallest multiple [tex]( n - n + 2)^{n-1}[/tex], but I still cannot express this as a function to the n power, and even if I did, the limit would just be 2.. so it's smaller than [tex]n![/tex], but not 'big enough'.

I think I may need a different approach. Suggestions?
 

Answers and Replies

  • #2
124
0
Would try to use [tex]r = \exp(\ln(r))[/tex]. This gives

[tex]\prod_{i = 1}^{n}\exp(\frac{\ln i}{n})[/tex]
 

Related Threads on Proof of lim (n!)^(1/n)

  • Last Post
Replies
13
Views
4K
  • Last Post
Replies
2
Views
1K
  • Last Post
Replies
6
Views
5K
  • Last Post
Replies
10
Views
1K
  • Last Post
Replies
6
Views
10K
  • Last Post
Replies
1
Views
1K
  • Last Post
Replies
18
Views
3K
  • Last Post
Replies
0
Views
10K
Replies
9
Views
10K
  • Last Post
Replies
4
Views
1K
Top