Sequence limit (factorial derivative?)

In summary: I just realized that there could be an uncountable infinity of values for n for which the inequality holds.In summary, the problem asks to find the limit of the sequence S_{n}=\frac{n^n}{n!}. The function either converges, approaches +∞ or -∞, or oscillates in such a way that it does not approach a definite value. To determine the limit, one must compare the growth rates of the numerator and denominator. Since the numerator grows faster, the limit is +∞. To formally prove this, one can use the fact that, for n>3, n^n>n!/n+1. This can be proven by induction. Therefore, the sequence diverges to +
  • #1
carlosbgois
68
0

Homework Statement



Find the limit of the sequence given by [itex]S_{n}=\frac{n^{n}}{n!}[/itex]

Homework Equations



[itex]lim_{n->∞}\frac{n^{n}}{n!}[/itex]

The Attempt at a Solution



I know the sequence diverges, but that doesn't mean the limit is also ∞, right?
 
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  • #2
carlosbgois said:

Homework Statement



Find the limit of the sequence given by [itex]S_{n}=\frac{n^{n}}{n!}[/itex]

Homework Equations



[itex]lim_{n->∞}\frac{n^{n}}{n!}[/itex]

The Attempt at a Solution



I know the sequence diverges, but that doesn't mean the limit is also ∞, right?

Either the function f(n) = n^n / n! converges, or else f(n) → +∞ or f(n) → -∞ or else f(n) "oscillates" as n → ∞ in such a way that f(n) does not approach a definite value---not even ± ∞. You need to decide which applies here.

RGV
 
  • #3
Ask yourself, which grows faster, the numerator or the denominator.
 
  • #4
Thank you both. As the numerator grows faster, and it's a divergence sequence, then the limit is +∞. Now, where may I start to formally prove it?
 
  • #5
Can you prove that [itex] n^n > n! [/itex] think about intervals.
 
  • #6
Yes, I can. For instance, I just evaluated [itex]lim_{n->∞}\frac{3^{n}}{(n+3)!}[/itex] as being 0 by showing that [itex]\forall x \geq 0, (n+3)!>3^{n}[/itex]. In a similar way, I may show that [itex]n^{n}>n![/itex] in the same interval.

It just seems to me that this method isn't rigorous enough, you know? As an example, 3>1 is true, but that does't mean that [itex]lim_{x->∞}\frac{x}{3x}=0[/itex]

Thank you
 
  • #7
You don't see that for arbitrary n

n times n > 1 times 2 times ... times n ?
 
  • #8
Yes, I do, but this seems as an intuitive approach, to me. Isn't it?

Just to be sure I got the concepts correctly: Let [itex]s_{n}=\frac{3^{n}}{(n+3)!}[/itex]. Then, [itex]s_{1}, s_{2}, ..., s_{n}[/itex] is a sequence, and the partial sum is [itex]S_{x}=s_{1}+s_{2}+...+s_{x}[/itex]. That being said, when I say I want to know the limit of the sequence [itex](lim_{n->∞}\frac{3^{n}}{(n+3)!})[/itex], I'm evaluating the "last" term, [itex]s_{n}[/itex], not the sum to the "last" term, [itex]S_{n}[/itex] right?

Many thanks
 
  • #9
Zondrina said:
Can you prove that [itex] n^n > n! [/itex] think about intervals.

This is not quite enough: you need [itex] n^n / n! [/itex] to be unbounded, not just > 1.

RGV
 
  • #10
Can you prove that, for n>3

[tex] n^n > \frac{n^n}{n!} > n +1 [/tex] ?
 
  • #11
dextercioby said:
Can you prove that, for n>3

[tex] n^n > \frac{n^n}{n!} > n +1 [/tex] ?

May it be done by induction? It clearly holds for n=3, then I assume it also holds for n=j, and show it's also valid for n=j+1. (Sorry, no paper and pen around right now, I'll try it as soon as I can)

Thanks
 
  • #12
I don't think my method for the second inequality (the 1st is obvious) is induction.
 

What is a sequence limit?

A sequence limit refers to the value that a sequence approaches as its index approaches infinity. It represents the maximum or minimum value that the terms of a sequence can reach.

What is a factorial derivative?

A factorial derivative is a type of derivative that is calculated using the factorial function. It is used to determine the rate of change of a function at a specific point by taking the ratio of the change in the function to the change in the factorial of the input variable.

How is the sequence limit related to the factorial derivative?

The sequence limit and factorial derivative are related through the concept of convergence. If a sequence has a limit, then its factorial derivative also has a limit. This is because the factorial derivative is a continuous function, meaning that it can be evaluated at any point on the sequence.

Can the sequence limit and factorial derivative have different values?

Yes, the sequence limit and factorial derivative can have different values. This is because the sequence limit represents the overall behavior of the sequence as its index approaches infinity, while the factorial derivative represents the instantaneous rate of change at a specific point.

How are sequence limits and factorial derivatives used in real life?

Sequence limits and factorial derivatives are used in various fields of science and mathematics, such as physics, engineering, and economics. They can be used to model and analyze complex systems and phenomena, such as population growth, chemical reactions, and stock market trends.

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