MHB Find a Sequence to Make lim(An/An+1)=∞

[esuahcdss12]

Hey

suppose I have sequence An

limAn,n→∞ = ∞

Is it possible to find a sequence which makes:

lim (An/An+1) ,n →∞ = ∞?

I tried to search a sequence like that and could not find, but I don't know how to prove that this is
can not be happening.
could you help please?

 

[S.G. Janssens]

I'm assuming you mean to study $\frac{A_n}{A_n + 1}$. Note that $\lim_{n \to \infty}\frac{A_n}{A_n + 1} = \lim_{n \to \infty}\frac{1}{1 + \frac{1}{A_n}}$.

 

[esuahcdss12]

I'm assuming you mean to study $\frac{A_n}{A_n + 1}$. Note that $\lim_{n \to \infty}\frac{A_n}{A_n + 1} = \lim_{n \to \infty}\frac{1}{1 + \frac{1}{A_n}}$.

no, I meant (A(n+1)/A(n))

 

[I like Serena]

Hi esuahcdss12!

So we want $\frac{A_{n+1}}{A_n}$ to diverge.
How about setting it for instance to be equal to $n$, which is the simplest expression that diverges.
That means we get $\frac{A_{n+1}}{A_n}=n\quad\Rightarrow\quad A_{n+1} = n A_n$.
Can we find a solution for that? (Wondering)

 

[esuahcdss12]

Hi esuahcdss12!

So we want $\frac{A_{n+1}}{A_n}$ to diverge.
How about setting it for instance to be equal to $n$, which is the simplest expression that diverges.
That means we get $\frac{A_{n+1}}{A_n}=n\quad\Rightarrow\quad A_{n+1} = n A_n$.
Can we find a solution for that? (Wondering)

It could be done only if $$n^n$$ and then after some calculation of $$\frac{A(n+1)}{An}$$
we get that the limit is e*lim(n+1) which is e *∞ = ∞
Is that correct?

 

[I like Serena]

It could be done only if $$n^n$$ and then after some calculation of $$\frac{A(n+1)}{An}$$
we get that the limit is e*lim(n+1) which is e *∞ = ∞
Is that correct?

Let's make that 'only if' with $Cn^n$ for some constant $C\ne 0$. Otherwise it's just an example (which it is anyways).
And yes, that is how the limit would be evaluated. (Nod)

 

[MarkFL]

Another example would be:

$$A_{n}=Cn!$$

So that:

$$\frac{A_{n+1}}{A_{n}}=\frac{C(n+1)!}{Cn!}=n+1$$

 

[I like Serena]

Another example would be:

$$A_{n}=Cn!$$

Oh wait! (Wait)
I think my example should be $A_n=Cn!$ instead of $A_n=Cn^n$. (Blush)
Then again $A_n=Cn^n$ also works.