What is the limit of (kn)! / n^(kn) as n approaches infinity?

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In summary, the conversation discusses the limit of a specific expression for any natural number k. For k=0, the limit is 1, and for k>0 it can be expressed using Stirling's approximation and is equal to 0 if k=1,2 and infinity if k>e. The conversation also mentions using the Gamma function to prove a limit involving infinite products.
  • #1
bomba923
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(This isn't homework, just a curiosity derived from another problem)

Well, this is probably quite simple...:shy:

For any natural 'k', what is the

[tex]\mathop {\lim }\limits_{n \to \infty } \frac{{\left( {kn} \right)!}}
{{n^{kn} }} [/tex]

?
 
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  • #2
I love it!

For k=0, it's 1. For k>0, we have

[tex]\mathop {\lim }\limits_{n \to \infty } \frac{{\left( {kn} \right)!}}{{n^{kn} }} = \sqrt{2\pi} \mathop {\lim }\limits_{n \to \infty } \frac{{\left( {kn} \right)}^{nk+\frac{1}{2}}e^{-nk}}{{n^{kn} }} = \sqrt{2k\pi} \mathop {\lim }\limits_{n \to \infty } \sqrt{n}\left( \frac{k}{e}\right) ^{nk} = \left\{\begin{array}{cc}0,&\mbox{ if }
k=1,2\\ \infty, & \mbox{ if } k>e\end{array}\right.[/tex]

since by Stirling's approximation: for [itex]n \gg 1[/itex],

[tex]n! \sim \sqrt{2\pi}n^{n+\frac{1}{2}}e^{-n}[/tex].
 
  • #3
Assuming you have knowledge of the Gamma function, try this one:

Prove that [itex]\forall n,k\in\mathbb{Z} ^{+}[/itex],

[tex]\mathop {\lim }\limits_{N \to \infty } \frac{\left[ \Gamma \left( 1+ \frac{k}{N}\right) \right] ^{n}}{\Gamma\left( 1+ \frac{nk}{N}\right)} =1[/tex]

Hint: Use infinite products!
 
  • #4
Thanks; when I mentioned "curiousity derived from another problem"
I was trying to find (from the product)

[tex]\mathop {\lim }\limits_{n \to \infty } \prod\limits_{i = 1}^{kn} {\frac{i}
{n}} = \mathop {\lim }\limits_{n \to \infty } \frac{{\left( {kn} \right)!}}
{{n^{kn} }} [/tex]
 

1. What is a limit in mathematics?

A limit in mathematics is a fundamental concept that describes the behavior of a function as its input approaches a certain value or "limit". It is used to determine the value that a function approaches rather than the value it takes at that specific point.

2. How do you calculate a limit?

To calculate a limit, you need to first identify the function and the value it is approaching. Then, you can use various techniques such as substitution, factoring, or the use of L'Hopital's rule to evaluate the limit. In some cases, you may need to use a graphing calculator or computer software to find the limit.

3. What is the purpose of finding a limit?

Finding a limit is important in mathematics because it helps us understand the behavior of a function and make predictions about its values. It is also used in various fields of science and engineering to model and analyze real-world phenomena.

4. Can a limit be undefined?

Yes, a limit can be undefined if the function has a vertical asymptote or a discontinuity at the value it is approaching. In these cases, the limit does not exist because the function does not approach a specific value, but instead approaches infinity or does not approach anything at all.

5. What are some real-life applications of limits?

Limits have many real-life applications, such as in physics to calculate the velocity of an object, in economics to analyze supply and demand, and in medicine to model the growth of tumors. They are also used in creating computer algorithms and in designing bridges and buildings.

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