Does the Sequence a_{n}=\frac{(n+2)!}{n!} Converge or Diverge?

[Winzer]

Homework Statement

Determine wheatehr the sequence diiverges or converges:

Homework Equations

a_{n}=\frac{(n+2)!}{n!}

The Attempt at a Solution

I was going to treat it using limits but the factorial is not defined for a function.
How do I deal with this?
Edit: sorry it is suppose to be: a_{n}=\frac{(n+2)!}{n!} nota_{n}=\frac{(n+1)!}{n!}

 

[CaptainZappo]

Try rewriting (n+1)! in a different form. What do you notice about the relationship between (n+1)! and n!?

 

[Winzer]

Ok I am pretty sure the it will diverge because (n+1)! goes faster to infinity than n!

rewrite like: a_{n}= 1+\frac{1}{n!} ?

 

[CaptainZappo]

Your rewritten version of the equation is not equal to the original equation. In fact, I believe the rewritten equation converges to 1 (do you see why?), thereby contradicting your statement above.

Please rewrite the numerator again, keeping in mind the goal of trying to simplify the equation.

 

[Winzer]

can i say n! = n(n-1)! and sub that in?

 

[CaptainZappo]

Yes! You are on the right track. If you try rewriting the numerator instead of the denominator, I think you will have more luck.

 

[Winzer]

but how would i rewrite the numerator?

 

[matt grime]

So, you know that n!= n(n-1)!... why is it not clear to you that that is precisely the same as (n+1)!=(n+1)n!? Or (n+1)!=(n+2)*(n+1)! ?

 

[Winzer]

how is n!= n(n-1)! the same as (n+1)!=(n+1)n!??

 

[Winzer]

what I meant was how did u get that?

 

[Dick]

what I meant was how did u get that?

The formula is true for all n. If n!=n*(n-1)! then changing n->n+1 gives (n+1)!=(n+1)*n!. In this sense, it's the 'same formula'. You may wish to try this for n->n+2. Can you show (n+2)!=(n+2)*(n+1)*n! at least for large enough n?

 

[Winzer]

Oh ok, thanks dick for clarifying, I just didn't think of it like that.
Indeed I worked it out and n->n+2 is (n+2)!=(n+2)*(n+1)*n!.