Calculating the Limit of n!^(1/n) and (1/n)(n!)^(1/n)

In summary, the problem is to calculate the limit for lim n!^{\frac{1}{n}} and lim {\frac{1}{n}} (n!)^{\frac{1}{n}}. The provided hints include the theorem: lim inf |\frac{Sn+1}{Sn}| \leq lim inf |Sn|^{\frac{1}{n}} \leq lim sup |Sn|^{\frac{1}{n}} \leq lim sup |{\frac{Sn+1}{Sn}}| and lim {(1+ {\frac{1}{n}})}^n = e. However, the individual is still unsure of how to solve the problem and is seeking help. They also mention the importance of
  • #1
jbear12
13
0
The problem is to caculate the limit for:
[tex]lim n!^{\frac{1}{n}}[/tex]

and
[tex]lim {\frac{1}{n}} (n!)^{\frac{1}{n}}[/tex]

Hints to this problem are
the theorem:
[tex]lim inf |\frac{Sn+1}{Sn}| \leq lim inf |Sn|^{\frac{1}{n}} \leq lim sup |Sn|^{\frac{1}{n}} \leq lim sup |{\frac{Sn+1}{Sn}}| [/tex]

and
[tex]lim {(1+ {\frac{1}{n}})}^n = e [/tex]

I really don't have a clue how to solve this, not even how to apply the hint.
Please help! Thanks!
 
Last edited:
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  • #2
Anyone?
 
  • #3
I don't hace an answer to this, but it might be best to learn how to use the forum LaTex editor, it's hard to read your syntax.

wrap your latex code with (tex) and (/tex) (replace the () with [] in your post to make it work), until you get used to using latex, just use the latex equation editor in your post, the latex reference button when posting your post.
 
  • #4
Hi Senjai
Can you tell me what's wrong with my code? I spent half an hour editing this and it only displays the first part...I have a midterm tomorrow :(
 
  • #5
I mean the hint theorem
 
  • #6
it works now :D
 
  • #7
anyone?
 

FAQ: Calculating the Limit of n!^(1/n) and (1/n)(n!)^(1/n)

1. What is the formula for calculating the limit of n!^(1/n)?

The formula for calculating the limit of n!^(1/n) is e, where e is the base of the natural logarithm. This means that as n approaches infinity, the value of n!^(1/n) will approach e.

2. How do I solve for the limit of (1/n)(n!)^(1/n)?

To solve for the limit of (1/n)(n!)^(1/n), we can use the same formula as for n!^(1/n), which is e. However, we must first manipulate the expression to be in the form of n!^(1/n). This can be done by taking the reciprocal of both sides, giving us the expression (n!)^(1/n)/n. We can then rewrite this as n!^(1/n) * (1/n), which is equivalent to the original expression. Therefore, the limit of (1/n)(n!)^(1/n) is also e.

3. What is the significance of the limit of n!^(1/n)?

The limit of n!^(1/n) is significant because it helps us understand the behavior of factorials as n approaches infinity. It shows that the value of n! becomes increasingly large as n increases, but at a decreasing rate. This means that the growth of n! slows down as n gets larger, and it eventually approaches a finite value of e.

4. Is there a graphical representation of the limit of n!^(1/n)?

Yes, there is a graphical representation of the limit of n!^(1/n). This can be seen by graphing the function y = n!^(1/n) and observing the shape of the curve as n approaches infinity. The curve will approach a horizontal asymptote at y = e, indicating that the limit of the function is e.

5. How can the limit of n!^(1/n) be used in real-world applications?

The limit of n!^(1/n) has various real-world applications, particularly in mathematics and statistics. It is used to calculate probabilities in certain scenarios, such as in the Poisson distribution. It is also used in the study of growth rates, as it shows the rate at which a factorial function grows as n increases. Additionally, it has applications in computer science, such as in the analysis of algorithms and sorting methods.

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