How Do I Prove That the Limit of \( n!^{n^{-n}} \) Equals 1?

  • Context: Graduate 
  • Thread starter Thread starter bomba923
  • Start date Start date
  • Tags Tags
    Limit
Click For Summary
SUMMARY

The limit of \( n!^{n^{-n}} \) as \( n \) approaches infinity equals 1. The discussion outlines the process of letting \( y = (n!)^{(n^{-n})} \) and applying the natural logarithm to both sides, resulting in \( \ln y = \frac{\ln(n!)}{n^n} \). By utilizing L'Hôpital's rule or recognizing that \( n^n \) grows faster than \( \ln(n!) \), the limit \( L \) can be determined. Finally, substituting back gives \( y = e^L \), confirming that the limit is indeed 1.

PREREQUISITES
  • Understanding of limits in calculus
  • Familiarity with factorial notation and properties
  • Knowledge of L'Hôpital's rule
  • Basic concepts of logarithms and exponential functions
NEXT STEPS
  • Study the application of L'Hôpital's rule in various limit problems
  • Explore Stirling's approximation for factorials
  • Learn about the growth rates of functions, particularly \( n! \) versus \( n^n \)
  • Investigate the properties of logarithmic functions in calculus
USEFUL FOR

Mathematicians, calculus students, and anyone interested in advanced limit evaluation techniques.

bomba923
Messages
759
Reaction score
0
How do I show that [tex]\mathop {\lim }\limits_{n \to \infty } \left( {n!} \right)^{\left( {n^{ - n} } \right)} = 1[/tex] ?

(The n^-n forces the value to decrease faster than n! increases, I believe. But how to work out that?)
 
Last edited:
Physics news on Phys.org
1)Firstly we'll let the expression (n!)^(n^-n)=y, then taking ln on both sides give,
lny=ln(n!)/n^n.
2) We'll then find the limit of the expression at the RHS using Le Hopital's rule or by observation that n^n increases faster than ln(n!). =)
3) After finding the limit, L, all we have to do is substitute back the value of y, which is e^L
 
Last edited:

Similar threads

Undergrad How does the ratio test fail and the root test succeed here?
  • · Replies 6 ·
Replies
6
Views
5K
Graduate Limit of ##i^\frac{1}{n}## as ##n \to \infty##
  • · Replies 14 ·
Replies
14
Views
2K
Undergrad Infinite Series - Divergence of 1/n question
  • · Replies 5 ·
Replies
5
Views
3K
Undergrad What is the limit of (a^n)/n for a>1?
  • · Replies 7 ·
Replies
7
Views
4K
Undergrad Express the limit as a definite integral
  • · Replies 16 ·
Replies
16
Views
5K
Undergrad How do a bunch of integrals make an n-simplex or an n-cube?
  • · Replies 2 ·
Replies
2
Views
2K
Graduate How to sum an infinite convergent series that has a term from the end
  • · Replies 15 ·
Replies
15
Views
3K
Undergrad Why prove Taylor's theorem with p(x)=q(x)−((b−a)^n/(b−x)^n)q(a)?
  • · Replies 9 ·
Replies
9
Views
3K
MHB Can the Series Sum Be Expressed as an Integral as N Approaches Infinity?
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad How Does the Ratio Test Determine Convergence in Power Series?
  • · Replies 19 ·
Replies
19
Views
2K