Analyze the existence of the infimum, minimum, supreme and maximum of the following sequence

Question

Analyze the existence of the infimum, minimum, supreme and maximum of the following set: $A=\{{a_n}:n \in\mathbb{N}\}$ where $a_n=P(n).b_n$ and $P(x)=1/3x^3-x^2-3x+10$ and $(b_n)_{n\in\mathbb{N}}$ is a sequence of positive terms that verifies:

$b_1=b_2=b_3$

$b_n<b_{n+1}$

If we observe the behavior of $P(n)$ we can obviously appreciate that it grows infinitely, then diverges, and even if it grows monotonously, we know that it won't have supreme and maximum.

But, what happens with $(b_n)_{n\in\mathbb{N}}$ (apart from its monotonically increasing)? Should we consider the cases where is bounded then is convergent so it will have a supreme, and the cases where is not?

And then, what we can say about $a_n$?

Answer 1

So far, you know that:

$b_n$ is monotonically increasing

$b_n>{0}$

$b_n.p(x)=a_n$

I would start by observing that $b_n$ can either diverge or converge (if it is bounded). Then, work a little bit with the polynomial (how does it behave when $n\longrightarrow\infty$ ?) to study what happens with $a_n$.

Hint

$p(3)=-17$