Sequence limit (factorial derivative?)

carlosbgois

1. The problem statement, all variables and given/known data

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

2. Relevant equations

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

3. The attempt at a solution

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

Ray Vickson

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

Zondrina

Ask yourself, which grows faster, the numerator or the denominator.

carlosbgois

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?

Zondrina

Can you prove that [itex] n^n > n! [/itex] think about intervals.

carlosbgois

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

dextercioby

You don't see that for arbitrary n

n times n > 1 times 2 times ... times n ?

carlosbgois

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

Ray Vickson

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

RGV

dextercioby

Can you prove that, for n>3

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

carlosbgois

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

dextercioby

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