Prove that: the sequence {$a_n$} diverges to plus or minus infinity if $$a_n=a_pn^p+a_{p-1}n^{p-1}+a_{p-2}n^{p-2}+\ldots+a_0$$ with $p$ in natural natural numbers, and $a_i$ in real numbers. Also given is that $a_p$ doesn't equal $0$.
This is an exercise I found in my Analysis 1 book. The chapter is mainly about convergence of sequences. This section in particular focuses on sequences that diverge to infinity.
I've tried a ratio test, that is $\lim\limits_{n \to \infty} |\frac{a_{n+1}}{a_n}|=\alpha$. It's a theorem in my book that if $\alpha>1$, then the $\lim\limits_{n \to \infty} |{a_n}|=+\infty$
However when I calculate this limit, which I already have trouble with, I find the limit to be equal to 1. My book states that if $\alpha=1$, then ${a_n}$ may converge, diverge to plus or minus infinity or oscilate. This information isn't really helpful of course.
So I was wondering if anyone here could help me?
Also note that this is an introductory course on Analysis, the second chapter. That means that there should probably be a rather 'easy' solution.
Thanks in advance.