Proof That $\lim_{n \rightarrow \infty} \frac{n}{(n!)^\frac{1}{n}} = e$

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In summary, the equation for the limit of n over the nth root of n factorial is $\lim_{n \rightarrow \infty} \frac{n}{(n!)^\frac{1}{n}} = e$. This limit can be proven using the squeeze theorem and the properties of limits. It is significant because it is a fundamental result in calculus and is used in many applications. The limit can also be generalized to other sequences using the same techniques and properties of limits. This limit is directly related to the mathematical constant e, as it is equal to e and was one of the first expressions used to define the constant.
  • #1
JonF
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does...

[tex] \lim_{n \rightarrow \infty} \frac{n}{(n!)^\frac{1}{n}} = e [/tex]

If not, is it divergent?
 
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  • #2
isnt there somethiong called stirlings formula for n! ?? Maybe you could use that and lhopital.
 
  • #3
well i just looked up stirling and it seems to suggest at a quick calculation, not guaranteed, that this limit is e/sqrt(2pi)
 
  • #4
e/(2pi)^(1/2) aprox= 1.0844

my calc can do the limit up to 200 and it equals about 2.67021... that's why i thought it may = e
 
  • #5
With Stirlings approximation: [itex]N!\approx N^Ne^{-N}[/itex], you indeed get:

[tex]\frac{N}{(N^Ne^{-N})^{\frac{1}{N}}}=\frac{N}{Ne^{-1}}=e[/tex]
 

FAQ: Proof That $\lim_{n \rightarrow \infty} \frac{n}{(n!)^\frac{1}{n}} = e$

What is the equation for the limit of n over the nth root of n factorial?

The equation for the limit is $\lim_{n \rightarrow \infty} \frac{n}{(n!)^\frac{1}{n}} = e$.

How can this limit be proven?

This limit can be proven using the squeeze theorem and the properties of limits.

What is the significance of this limit?

This limit is significant because it is a fundamental result in calculus and is used in many applications, such as in the calculation of compound interest and in the derivation of the natural logarithm function.

Can this limit be generalized to other sequences?

Yes, this limit can be generalized to other sequences using the same techniques and properties of limits.

How does this limit relate to the mathematical constant e?

This limit is directly related to the mathematical constant e, as it is equal to e and was one of the first expressions used to define the constant.

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